Mathematics 241A

Introduction to Global Analysis

John Douglas MooreDepartment of Mathematics

University of CaliforniaSanta Barbara, CA, USA 93106e-mail: moore@math.ucsb.edu

Fall, 2010

Preface

These are slightly revised lecture notes for the graduate course, Topics in Ge-ometry, given at UCSB during the fall quarter of 2009. We intend to includeadditional topics later. It might be helpful to start with an overview of thesubject presented.

Morse theory might have developed in three main stages, although as eventstranspired, the three stages were actually intertwined.

The first stage should have been finite-dimensional Morse theory, which re-lates critical points of proper nonnegative functions on finite-dimensional man-ifolds to the topology of these manifolds. Indeed, the foundations for this werelaid in Marston Morse’s first landmark article on Morse theory [55], but hequickly turned his attention to problems from the calculus of variations, whichultimately became part of the infinite-dimensional theory. In subsequent devel-opments, finite-dimensional Morse theory became a primary tool for studyingthe topology of finite-dimensional manifolds and has had many successes, in-cluding the celebrated h-cobordism theorem of Smale [51], which settled thegeneralized Poincare conjecture in dimensions greater than five: any compactmanifold of dimension at least five which is homotopy equivalent to a spheremust be homeomorphic to a sphere. Modern expositions of finite-dimensionalMorse theory often construct a chain complex from the free abelian group gener-ated by the critical points, the boundary being defined by orbits of the gradientflow which connect the critical points. The homology of this chain complex,called Morse homology, is isomorphic to the usual integer homology of the man-ifold; see [74] for example.

What might have been the second stage—the Morse theory of geodesics—formed the core of what Morse [56] called “the calculus of variations in thelarge.” Morse was interested in studying the critical points of the length func-tion or action function on the space of paths joining two points in a Riemannianmanifold, the critical points being geodesics. His idea was to approximate theinfinite-dimensional space of paths by a finite-dimensional manifold of very highdimension, and then apply finite-dimensional Morse theory to this approxima-tion. As explained in Milnor’s classical book on Morse theory [50], this approachproduced many striking results in the theory of geodesics in Riemannian geome-try, such as the theorem of Serre that any two points on a compact Riemannianmanifold can be joined by infinitely many geodesics. It also provided the firstproof of the Bott periodicity theorem from homotopy theory. One might regard

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the Morse theory of geodesics as an application of topology to the study ofordinary differential equations, in particular, to those equations which like theequation for geodesics, arise from classical mechanics. Thus Morse theory arisesfrom the very core of “applied mathematics.”

Palais and Smale were able to provide an elegant reformulation of the Morsetheory of geodesics in the language of infinite-dimensional Hilbert manifolds[62]. They showed that the action function on the infinite-dimensional manifoldof paths satisfies “Condition C,” a condition replacing “proper” in the finite-dimensional theory, which they showed is sufficient for the development of Morsetheory in infinite dimensions. This became the standard approach to the Morsetheory of geodesics during the last several decades of the twentieth century.

Future historians will most likely regard the third stage of Morse theoryas encompassing many strands, but our viewpoint is to focus on techniquesfor applying Morse theory to nonlinear elliptic partial differential equationscoming from the calculus of variations in which the domain is a two-dimensionalcompact surface. Morse himself hoped to apply the ideas of his theory to acentral case—the partial differential equations for minimal surfaces in Euclideanspace. The first steps in this direction were taken by Morse and Tompkins, aswell as Shiffman, who established the theorem that if a simple closed curve inEuclidean space R3 bounds two stable minimal disks, it bounds a third, which isnot stable. This provided a version of the so-called “mountain pass lemma” forminimal disks in Euclidean space. The results of Morse, Tompkins and Shiffmansuggested that Morse inequalities should hold for minimal surfaces in Euclideanspace with boundary constrained to lie on a given Jordan curve, and indeed,such inequalities were later established under appropriate hypotheses [41].

But the most natural extension of the Morse theory of geodesics to the realmof partial differential equations would be a Morse theory of two-dimensionalminimal surfaces in a more general curved ambient Riemannian manifold M ,instead of the ambient Euclidean space used in the classical theory of minimalsurfaces. The generalization to a completely general ambient space requiresnew techniques. Unfortunately, if Σ is a connected compact surface, it becomessomewhat awkward to extend the finite-dimensional approximation procedure—so effective in the theory of geodesics—to the space of mappings Map(Σ,M)from Σ to M . One might hope that a better approach would be based upon thetheory of infinite-dimensional manifolds, as developed by Palais and Smale, buta serious problem is encountered: the standard energy function

E : Map(Σ,M)→ R,

used in the theory of harmonic maps and parametrized minimal surfaces, fails tosatisfy the Condition C which Palais and Smale had used so effectively in theirtheory, when Map(Σ,M) is completed with respect to a norm strong enough tolie within the space of continuous functions.

To get around this difficulty, Sacks and Uhlenbeck introduced the α-energy[68], [69], a perturbation of the usual energy which does satisfy Condition Cwhen Map(Σ,M) is completed with respect to a Banach space norm which

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is both weak enough to satisfy condition C and strong enough to have thesame homotopy type as the space of continuous maps from Σ to M . In simpleterms, we could say that the α-energy lies within “Sobolev range.” The α-energy approaches the usual energy (plus a constant) as the parameter α inthe perturbation goes to one, and we can therefore say that the usual energyis “on the border of Sobolev range.” Using the α-energy, Sacks and Uhlenbeckwere able to establish many striking results in the theory of minimal surfaces inRiemannian manifolds, including the fact that any compact simply connectedRiemannian manifold contains at least one nonconstant minimal two-sphere,which parallels the classical theorem of Fet and Lyusternik stating that anycompact Riemannian manifold contains at least one smooth closed geodesic. Butthey also discovered the phenomenon of “bubbling” as α → 1, which preventsMorse inequalitites from holding for compact parametrized minimal surfaces incomplete generality.

A somewhat different approach to existence of parametrized minimal sur-faces in Riemannian manifolds was developed at about the same time by Schoenand Yau [72], using Morrey’s solution to the Plateau problem in a Riemannianmanifold and arguments based upon a “replacement procedure.” Their approachyielded striking theorems relating positive scalar curvature to topology of three-manifolds, and was a step toward the first proof of the positive mass theorem ofgeneral relativity. The Schoen-Yau replacement procedure can also be used toobtain many of the existence results of Sacks and Uhlenbeck, and an alternatetreatment of many of their theorems is provided by Jost [39], as well as by otherauthors, who use yet different techniques, including heat flow.

However, the approach via Sacks-Uhlenbeck perturbation seems to providethe strongest link with the global analysis approach to nonlinear partial differ-ential equations, and the clearest insight into bubbling, the phenomenon men-tioned above which is observed as the perturbation is turned off. Bubblingappears to be an essential component of any complete critical point theory ofminimal surfaces within Riemannian manifolds, and this phenomenon also ap-pears in the study of other nonlinear partial differential equations, such as theYang-Mills equation on four-dimensional manifolds.

If one were to replace the Riemannian metric on the ambient manifold by aconformal equivalence class of such metrics, the theory of two-dimensional min-imal surfaces would become part of a broader context—nonlinear partial differ-ential equations which are conformally invariant and lie on the border of Sobolevrange. These equations include the Yang-Mills equations on four-dimensionalmanifolds from the standard model for particle physics, the anti-self-dual equa-tions used so effectively by Donaldson, the Seiberg-Witten equations, and Gro-mov’s equations for pseudoholomorphic curves. A technology has been devel-oped for studying these equations. First one needs to develop a transversalitytheory using Smale’s generalization of Sard’s theorem from finite-dimensionaldifferential topology. This generally shows that in generic situtations solutionsto the nonlinear partial differential equation form a finite-dimensional submani-fold of an infinite-dimensional function space. The tangent space to this subman-ifold is studied via the linearization of the nonlinear partial differential equation

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at a given solution; often, the dimension of the tangent space is obtained byapplication of the Atiyah-Singer index theorem, which reduces to the Riemann-Roch theorem in the case of parametrized minimal surfaces. Next one developsa suitable compactness theorem. Finally, one uses topological and geometricmethods to derive important geometrical conclusions (for example, existence ofminimal surfaces) or to construct differential topological invariants of manifolds(as in Seiberg-Witten theory).

As mentioned before, for minimal surfaces, bubbling implies that the mostobvious extension of the Morse inequalities to minimal surfaces in Riemannianmanifolds cannot hold, but also provides the framework for analyzing how theMorse inequalities fail.

Not only does Sacks-Uhlenbeck bubbling interfere with the Morse inequali-ties, but when minimizing area one must allow for variations not only over thespace of functions but over the conformal structure on the surface, an elementof Teichmuller space, and a sequence of harmonic maps may degenerate as theconformal structure moves to the boundary. Moreover, branched coverings ofa given minimal surfaces count as new critical points within the space of func-tions, although they are not geometrically distinct from the covered surface.Finally, the energy is invariant under the action of the mapping class group, sothe energy descends to a function on the quotient space, a space which has amore complicated topology than that of Map(Σ,M) when the genus of Σ is atleast two. One might suspect that a procedure for constructing minimal surfacesthat can fail in several different ways is too flawed to be of much use. However,we argue that a different perspective is more productive—since the minimaxprocedure for a given homology class does not always yield interesting minimalsurfaces, one should divide homology classes into various categories dependingupon which of the possible difficulties will likely arise.

It is our purpose here to provide some of the foundations for such a theory,a theory which promises important applications similar to those found in theMorse theory of geodesics.

It has long been known that that the extension of Morse theory to infinite-dimensional manifolds is not really necessary for the study of geodesics. Indeed,Bott expresses it this way in his beautiful survey article on Morse theory [9]in 1982: ”I know of no aspect of the geodesic question where [the infinite-dimensional approach] is essential; however it clearly has some aesthetic advan-tages, and points the way for situations where finite dimensional approximationsare not possible...” One could imagine constructing a finite-dimensional approx-imation suitable for studying the α-energy when α > 1, but it would be far moreawkward than the infinite-dimensional manifold of maps, and an analysis of howit breaks down as α→ 1 seems to require a study of infinite-dimensional mani-fold pieces. Thus, for any partial Morse theory of minimal surfaces, in contrastto closed geodesics, calculus on infinite-dimensional manifolds appears to playan essential conceptual and simplifying role.

We assume the reader is familiar with the basics of finite-dimensional dif-ferential geometry, including geodesics, curvature, the tubular neighborhoodtheorem, and the second variation formula for geodesics. We will also assume

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some familiarity with basic complex analysis, the foundations of Banach andHilbert space theory, and the willingness to accept results from the linear the-ory of elliptic partial differential operators, in particular the theory of Fredholmoperators on Sobolev spaces. All of these topics are treated very well in highlyaccessible sources, to which we can refer at the appropriate time.

We begin with an overview of global analysis on infinite-dimensional man-ifolds of maps. The second chapter reviews the theory of geodesics on Rie-mannian manifolds which owes much to the pioneering work of Bott, Gromoll,Klingenberg and Meyer. We then turn to minimal surfaces, providing a briefintroduction to the key theorems of Sacks, Uhlenbeck, Meeks, Schoen and Yauwhich helped elucidate the topology of three-dimensional manifolds. A finalchapter is planned that will cover the bumpy metric theorem of [52] and someof its immediate applications. Starred sections can be omitted without loss ofcontinuity.

Doug Moore

Santa Barbara, CA, December, 2010

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Contents

1 Infinite-dimensional manifolds 11.1 A global setting for nonlinear DE’s . . . . . . . . . . . . . . . . . 11.2 Infinite-dimensional calculus . . . . . . . . . . . . . . . . . . . . . 21.3 Infinite-dimensional manifolds . . . . . . . . . . . . . . . . . . . . 151.4 The basic mapping spaces . . . . . . . . . . . . . . . . . . . . . . 231.5 Homotopy type of the space of maps . . . . . . . . . . . . . . . . 291.6 The α-Lemma* . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.7 The tangent and cotangent bundles . . . . . . . . . . . . . . . . . 361.8 Differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.9 Riemannian and Finsler metrics . . . . . . . . . . . . . . . . . . . 431.10 Vector fields and ODE’s . . . . . . . . . . . . . . . . . . . . . . . 471.11 Condition C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491.12 Topological constraints give critical points . . . . . . . . . . . . . 531.13 de Rham cohomology . . . . . . . . . . . . . . . . . . . . . . . . 56

2 Morse Theory of Geodesics 632.1 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.2 Condition C for the action . . . . . . . . . . . . . . . . . . . . . . 662.3 Existence of smooth closed geodesics . . . . . . . . . . . . . . . . 712.4 Second variation . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.5 Morse nondegenerate critical points . . . . . . . . . . . . . . . . . 782.6 The Sard-Smale Theorem . . . . . . . . . . . . . . . . . . . . . . 822.7 Existence of Morse functions . . . . . . . . . . . . . . . . . . . . 852.8 Bumpy metrics for smooth closed geodesics* . . . . . . . . . . . . 892.9 Adding handles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 962.10 Morse inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 1002.11 The Morse-Witten complex . . . . . . . . . . . . . . . . . . . . . 104

3 Harmonic and minimal surfaces 1103.1 The energy of a smooth map . . . . . . . . . . . . . . . . . . . . 1103.2 Minimal surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163.3 Minimal surfaces of higher genus . . . . . . . . . . . . . . . . . . 1223.4 The Bochner Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 1293.5 The α-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

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3.6 Regularity of (α, ω)-harmonic maps . . . . . . . . . . . . . . . . . 1383.7 Morse theory for the perturbed energy . . . . . . . . . . . . . . . 1413.8 Local control of energy density . . . . . . . . . . . . . . . . . . . 1473.9 Bubbling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1533.10 Existence of minimizing spheres . . . . . . . . . . . . . . . . . . . 1563.11 Existence of minimal tori . . . . . . . . . . . . . . . . . . . . . . 1633.12 Higher genus minimal surfaces* . . . . . . . . . . . . . . . . . . . 1663.13 Complex form of second variation . . . . . . . . . . . . . . . . . . 1683.14 Isotropic curvature . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Bibliography 176

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Chapter 1

Infinite-dimensionalmanifolds

1.1 A global setting for nonlinear DE’s

Linear differential equations are often fruitfully studied via techniques from lin-ear functional analysis, including the theory of Banach and Hilbert spaces. Incontrast, the proper setting for an important class of nonlinear partial differen-tial equations is a nonlinear version of functional analysis, which is based uponinfinite-dimensional manifolds modeled on Banach and Hilbert spaces. The the-ory of such manifolds was developed by Eells, Palais and Smale among others inthe 1950’s and 1960’s, and has proven to be extremely useful for understandingmany of the nonlinear differential equations which arise in geometry, such as

1. the equation for geodesics in a Riemannian manifold,

2. the equation for harmonic maps from surfaces into a Riemannian manifold,or for minimal surfaces in a Riemannian manifold,

3. the equations for pseudoholomorphic curves in a symplectic manifold,

4. and the Seiberg-Witten equations.

In all of these examples, the solutions can be expressed as critical points ofa real-valued function (often called the action or the energy) defined on aninfinite-dimensional manifold, such as the function space manifolds describedin the following pages. In favorable cases, a gradient (or pseudogradient) ofthe action or energy can then be used to locate critical points (solutions to thedifferential equations) via what is often called the “method of steepest descent.”

For this procedure to work, the topology on the function space must bestrong enough for the action or energy to be differentiable, yet weak enough toforce convergence of a sequence which is tending toward a minimum (or to aminimax solution for a given constraint). The two conflicting conditions often

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select a unique acceptable topology for the space of functions. In the mostfavorable circumstances, the topology is strong enough so that it lies withinthe space of continuous functions, a space which has been studied extensivelyby topologists. The function space is then often homotopy equivalent, that isequivalent in the sense of homotopy theory, to the space of continuous functions.This makes some existence questions within the theory of nonlinear differentialequations accessible by topological methods.

In the case of the Seiberg-Witten equations, the logic is reversed. Instead oftopology shedding light on existence of solutions to partial differential equations,it is the space of solutions to the Seiberg-Witten equations that enable one todistinguish between different smooth structures on smooth four-manifolds (asexplained in [54]). This illustrates that at a very fundamental level, topologyand nonlinear partial differential equations are closely related, and underscoresthe importance of developing a global theory of partial differential equationsbased upon the theory of infinite-dimensional manifolds.

1.2 Infinite-dimensional calculus

Our first topic is the theory of infinite-dimensional manifolds. We refer to theexcellent presentations of Lang [43] or of Abraham, Marsden and Ratiu [2] forfurther elaboration of topics only briefly introduced in the following pages.

It was pointed out by Smale, Abraham, Lang and others in the 1960’s thatseveral variable calculus could be developed not just in finite-dimensional Eu-clidean spaces, but also with very little extra work within the context of infinite-dimensional Hilbert and Banach spaces. Most of the proofs of theorems arestraightforward modifications of the proofs in Rn, so we will go very rapidlyover this basic material.

Definition. A pre-Hilbert space is a real vector space E together with a function〈·, ·〉 : E × E → R which satisfies the following axioms:

1. 〈x, y〉 = 〈y, x〉, for x, y ∈ E.

2. 〈ax, y〉 = a〈x, y〉, for a ∈ R and x, y ∈ E.

3. 〈x+ y, z〉 = 〈x, z〉+ 〈y, z〉, for x, y, z ∈ E.

4. 〈x, x〉 ≥ 0, for x ∈ E, equality holding only when x = 0.

These axioms simply state that 〈x, y〉 is a positive-definite symmetric bilinearform on E.

Given such a pre-Hilbert space, we can define a map ‖ · ‖ : E → R by‖x‖ =

√〈x, x〉. We can regard E as a metric space by defining the distance

between elements x and y of E to be d(x, y) = ‖x− y‖.

Definition. A Hilbert space is a pre-Hilbert space which is complete in termsof the metric d.

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An example of a finite-dimensional Hilbert spaces is Rn with inner product 〈·, ·〉defined by

〈(x1, . . . , xn), (y1, . . . yn)〉 = x1y1 + · · ·+ xnyn.

An example of an infinite dimensional Hilbert space is the space L2([0, 1],R)studied in basic analysis courses. To construct it, one starts with defining aninner product 〈·, ·〉 on the space

C∞([0, 1],R) = C∞ functions f : [0, 1]→ R

by

〈φ, ψ〉 =∫ 1

0

φ(t)ψ(t)dt.

This inner product satisfies the first four of the above axioms but not the lastone. We let L2([0, 1],R) denote the equivalence classes of Cauchy sequencesfrom C∞([0, 1],R), two Cauchy sequences φi and ψi being equivalent if foreach ε > 0 there is a positive integer N such that

i, j > N ⇒ ‖φi − ψi‖ < ε. (1.1)

Equivalence classes of sequences form a vector space, and we define an innerproduct 〈·, ·〉 on L2([0, 1],R) by

〈φi, ψi〉 = limi,i→∞

〈φi, ψi〉.

We say that L2([0, 1],R) is the completion of C∞([0, 1],R) with respect to 〈·, ·〉.The process we have described is virtually the same as that often used to con-struct the real numbers from the rationals.

Definition. A pre-Banach space is a vector space E together with a function‖ · ‖ : E → R which satisfies the following axioms:

1. ‖ax‖ = |a|‖x‖, when a ∈ R and x ∈ E.

2. ‖x+ y‖ ≤ ‖x‖+ ‖y‖, when x, y ∈ E.

3. ‖x‖ ≥ 0, for x ∈ E

4. ‖x‖ = 0 only if x = 0.

A function ‖ · ‖ : E → R which satisfies the first three axioms is called aseminorm on E. If, in addition, it satisfies the fourth axiom it is called a norm.

As in the case of Hilbert spaces, we can make E into a metric space bydefining the distance between elements x and y of E to be d(x, y) = ‖x− y‖.

Definition. A Banach space is a pre-Banach space which is complete in termsof the metric d.

Of course, every Hilbert space is a Banach space with norm ‖ · ‖ defined by‖x‖ =

√〈x, x〉. Let

C0([0, 1],R) = continuous functions f : [0, 1]→ R ,

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and define

‖ · ‖ : C0([0, 1],R)→ R by ‖f‖ = sup|f(t)| : t ∈ [0, 1].

Then ‖ · ‖ makes C0([0, 1],R) into a Banach space. More generally, we canconsider the space

Ck([0, 1],R) = functions f : [0, 1]→ R whichhave continuous derivatives up to order k ,

a Banach space with respect to the norm

‖ · ‖k : Ck([0, 1],R)→ R defined by ‖f‖k = sup

k∑i=0

|f (i)(t)| : t ∈ [0, 1]

,

(1.2)where f (i)(t) denotes the derivative of f of order i.

When 1 ≤ p < ∞ and p 6= 2, the spaces Lp([0, 1],R) studied in basicanalysis courses are Banach spaces which are not Hilbert spaces. To constructthese spaces, one starts with defining a function ‖ · ‖ on the space C∞([0, 1],R)of C∞ functions f : [0, 1]→ R by

‖φ‖ =[∫ 1

0

|φ(t)|pdt]1/p

,

which is shown to be a norm by means of the Minkowski inequality. This agreeswith the norm previously defined on L2([0, 1],R) when p = 2. Just as in theconstruction of L2([0, 1],R), one can use this norm to define Cauchy sequences,and let Lp([0, 1],R) be the set of equivalence classes of Cauchy sequences, wheretwo Cauchy sequences φi and ψi are equivalent if for each ε > 0 there is apositive integer N such that (1.1) holds. The basic properties of Lp spaces aretreated in standard references on functional analysis; thus for the Holder andMinkowski inequalities, for example, one can refer to Theorem III.1 of [65].

Each Banach space E has a metric

d : E × E → R defined by d(e1, e2) = ‖e1 − e2‖,

and we say that a subset U of E is open if

p ∈ U ⇒ Bε(p) = q ∈ E : d(p.q) < ε ⊂ U,

for some ε > 0. A map T : E1 → E2 between Banach spaces is continuous ifT−1(U) is open for each open U ⊂ E2, or equivalently, if there is a constantc > 0 such that

‖T (e1)‖2 ≤ c‖e1‖1, for all e1 ∈ E1,

where ‖ · ‖1 and ‖ · ‖2 are the norms on E1 and E2 respectively.

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We let L(E1, E2) be the space of continuous linear maps T : E1 → E2, aBanach space in its own right under the norm

‖T‖ = sup‖T (e1)‖ : e1 ∈ E1, ‖e1‖ = 1.

It is easily shown that L(E1, E2) is the same as the space of linear maps fromE1 to E2, which are bounded in terms of this norm. In particular, we can definethe dual of a Banach space E to be E? = L(E,R). We say that a Banach spaceis reflexive if (E?)? is isomorphic to E. It is proven in analysis courses thatLp([0, 1],R) is reflexive when 1 < p < ∞ but L1([0, 1],R) and C0([0, 1],R) arenot.

Banach spaces and continuous linear maps form a category, as do Hilbertspaces and continuous linear maps. The subject functional analysis is con-cerned with properties of the categories of Hilbert spaces, Banach spaces andmore general spaces of functions, and is one of the major tools in studying linearpartial differential equations. Key theorems from the theory of Banach spacesinclude the Open Mapping Theorem, the Hahn-Banach Theorem and the Uni-form Boundedness Theorem. These theorems are discussed in [65], [66] and [67];we will need to use the statements of these theorems and their consequences.

The Open Mapping Theorem states that if T : E1 → E2 is a continuoussurjective map between Banach spaces, it takes open sets to open sets. Thus if Tis a continuous bijection, its inverse is continuous. The Hahn-Banach Theoremimplies that if e is a nonzero element of a Banach space E, then there is acontinuous linear function λ : E → R such that λ(e) 6= 0. A bilinear mapB : E1 ×E2 → F is said to be continuous if there is a constant c > 0 such that

‖B(e1, e2)‖ ≤ c‖e1‖1‖e2‖1, for all e1 ∈ E1 and all e2 ∈ E2.

One of the consequences of the Uniform Boundedness Theorem is that such aBilinear map is continuous is and only if

B(·, e2) : E1 → F and B(e1, ·) : E1 → F

are continuous for each e1 ∈ E1 and each e2 ∈ E2. The three theorems arealmost trivial to prove for finite-dimensional Banach spaces, but the proofs aremore subtle for infinite-dimensional Banach spaces.

We now turn to the question of how to develop differential calculus for func-tions defined on Banach spaces. It is actually the topology, or the equivalenceclass of norms on the Banach space, that is important for calculus, two norms‖ · ‖1 and ‖ · ‖2 on a linear space E being equivalent if there is a constant c > 1such that

1c‖x‖1 < ‖x‖2 < c‖x‖1, for x ∈ E.

Lang [43] calls a vector space E a Banachable space if it is endowed with anequivalence class of Banach space norms. However, most other authors do notuse this term, and we will simply use the simpler term Banach space, it beingunderstood however, that we may sometimes pass to an equivalent norm in the

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middle of an argument, when it is the underlying vector space with its topology,the “topological vector space”—not the norm itself—that is important.

Definition. Suppose that E1 and E2 are Banach spaces, and that U is an opensubset of E1. A continuous map f : U → E2 is said to be differentiable at thepoint x0 ∈ U if there exists a continuous linear map T : E1 → E2 such that

lim‖h‖→0

‖f(x0 + h)− f(x0)− T (h)‖‖h‖

= 0,

where ‖ · ‖ denotes both the Banach space norms on E1 and E2. We will call Tthe derivative of f at x0 and write Df(x0) for T . Note that the derivative canalso be calculated from the formula

Df(x0)h = limt→0

f(x0 + th)− f(x)t

.

Just as in ordinary calculus, the derivative Df(x0) determines the linearizationof f near x0, which is the affine function

f(x) = f(x0) +Df(x0)(x− x0)

which most closely approximates f near x0.If f is differentiable at every x ∈ U , a derivative Df(x) is defined for each

x ∈ U and thus we have a set-theoretic map Df : U → L(E1, E2). If this mapDf is continuous, we can also ask whether it is differentiable at x0 ∈ U . Thiswill be the case if there is a continuous linear map T : E1 → L(E1, E2) suchthat

lim‖h‖→0

‖Df(x0 + h)−Df(x0)− T (h)‖‖h‖

= 0,

in which case we write D2f(x0) for T and call D2f(x0) the second derivative off at x0. Note that

D2f(x0) ∈ L(E1, L(E1, E2)) = L2(E1, E2)= continuous bilinear maps T : E1 × E1 → E2,

which can also be made into a Banach space in an obvious way.We say that a function f : U → E2, where U is an open subset of E1, is

1. C0 if it is continuous.

2. C1 if it is continuous and differentiable at every x ∈ U , and Df : U →L(E1, E2) is continuous.

3. Ck for k ≥ 2 if it is C1 and Df : U → L(E1, E2) is Ck−1.

4. C∞ or smooth if it is Ck for every nonnegative integer k.

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With the above definition of differentiation, many of the arguments in severalvariable calculus can be transported without difficulty to the Banach spacesetting, as carried out in detail in [43] or [2]. For example, the Leibniz rule fordifferentiating a product carries over immediately to the infinite-dimensionalsetting:.

Proposition 1.2.1. Suppose that B : F1 × F2 → G is a continuous bilinearmap between Banach spaces, that U is an open subsets of a Banach space Eand

f : U1 −→ F1, f2 : U2 −→ F2

are C1 maps. Then e 7→ g(e) = B(f1(e), f2(e)): is a C1 map, and

Dg(x0)h = B(Df1(x0)h, f2(e)) +B(f1(x0)Df2(x0)h).

Sketch of Proof: To simplify notation, we write

g(x) = B(f1(x), f2(x)) = f1(x) · f2(x).

Then

g(x+ h)− g(x) = (f1(x+ h)− f1(x))f2(x+ h) + f1(x)(f2(x+ h)− f2(x)),

and hence

g(x+ h)− g(x)‖h‖

=f1(x+ h)− f1(x)

‖h‖f2(x+ h) + f1(x)

f2(x+ h)− f2(x)‖h‖

.

The Proposition follows by taking the limit as ‖h‖ → 0.

The following lemma is called the chain rule.

Proposition 1.2.2. If U1 and U2 are open subsets of Banach spaces E1 andE2 and

f : U1 −→ U2, g : U2 −→ E3

are C1 maps, then so is g f : U1 → E3, and

D(g f)(x0) = Dg(f(x0))Df(x0), for x0 ∈ U1.

Sketch of Proof: We have

f(x0 + h) = f(x0) +Df(x0)h+ o(h),

where the symbol o(h) stands for an element in E2 such that o(h)/‖h‖ → 0 as‖h‖ → 0. Similarly,

g(f(x0) + k) = g(f(x0)) +Dg(f(x0))(k) + o(k).

7

Setting k = Df(x0)h+ o(h) yields

g(f(x0 + h)) = g(f(x0)) +Dg(f(x0))(Df(x0)h+ o(h)) + o(k).

One checks without difficulty using continuity of g that an o(k) expression,where k is a bounded linear function of h, is also o(h), and hence

g(f(x0 + h)) = g(f(x0)) +Dg(f(x0))Df(x0)h+ o(h),

which gives the desired conclusion.

By induction, one immediately shows that the composition of Ck maps is Ck

and the composition of C∞ maps is C∞.

Example 1.2.3. We suppose that the domain of the function is the Banachspace E = Lp(S1,RN ), the completion of the space C∞(S1,RN ) of smoothRN -valued functions on S1 with respect to the Lp-norm

‖φ‖Lp =[∫

S1|φ(t)|pdt

]1/p

, for φ ∈ Lp(S1,RN ).

Here S1 is regarded as the quotient of the interval [0, 1] obtained by identifyingthe points 0 and 1, and possesses the standard measure dt with respect to whichS1 has measure one. A useful tool for dealing with functions in the Lp spacesis the Holder inequality which states: if φ ∈ Lp, ψ ∈ Lq and

1p

+1q

=1r

where p, q, r ≥ 1, then φψ ∈ Lr and ‖φψ‖Lr ≤ ‖φ‖Lp‖ψ‖Lq .

Using this inequality and the chain rule, it is not difficult to show that whenp ≥ 2, the function

f : E −→ R defined by f(φ) =∫S1

(1 + |φ(t)|2)p/2dt

is C2 and that its first and second derivatives are given by the formulae

Df(φ)(ψ) = p

∫S1

(1 + |φ(t)|2)p/2−1φ(t) · ψ(t)dt

and

(D2f)(φ)(ψ1, ψ2) = p

∫S1

(1 + |φ(t)|2)p/2−1ψ1(t) · ψ2(t)dt

+ p(p− 2)∫S1

(1 + |φ(t)|2)p/2−2(φ(t) · ψ1(t))(φ(t) · ψ2(t))dt.

Indeed, to carry this out, we apply the chain rule to f = h g, where h isintegration over S1 and

g : E → L1(S1,R) by g(φ) = (1 + |φ(t)|2)p/2.

8

On the other hand, if f were C3, one can verify that g would also be C3, withthird derivative given by the formula

(D3g)(ψ1, ψ2, ψ3) = (3p/2)(p/2− 1)(1 + |φ|2)p/2−1φψ1ψ2ψ3

+ (3p/2)(p/2− 1)(p/2− 2)(1 + |φ|2)p/2−2φ3ψ1ψ2ψ3,

and if p < 3, for any smooth choice of φ, we could choose ψ1, ψ2 and ψ3 in Lp

such that the product ψ1ψ2ψ3 is not in L1. This implies that f cannot be C3

when 2 < p < 3.Another familiar theorem from several variable calculus in finite dimension

is the “equality of mixed partials.” To state the infinite-dimensional version, welet E and F be Banach spaces, U an open subset of E. Suppose that f : U → Fis a C2 map. Then for x0 ∈ U ,

D2f(x0) ∈ L(E,L(E,F )) = L2(E,F )= continuous bilinear maps T : E × E → F.

Of course, a very important case is the one where F = R, the base field of realnumbers.

Proposition 1.2.4. D2f(x0) is symmetric; in other words,

D2f(x0)(h, k) = D2f(x0)(k, h), for h, k ∈ E.

Sketch of proof: First note that by the Hahn-Banach theorem, it suffices toshow that if λ : F → R is any continuous linear functional, then D2(λf)(x0) =λ D2f(x0) is symmetric, because if D2f(x0) is not symmetric, the same willbe true of λ D2f(x0), for some linear function λ. This reduces the proof tothe case where F = R. Next note that via the chain rule,

f(x+ h)− f(x) =∫ 1

0

(Df)(x+ th)hdt,

and by iteration,

f(x+ h+ k)− f(x+ k)− f(x+ h) + f(x) =∫ 1

0

(Df)(x+ th+ k)hdt

=∫ 1

0

∫ 1

0

(D(Df)(x+ th+ sk)(k))hdsdt.

Interchanging h and k yields

f(x+ h+ k)− f(x+ h)− f(x+ k) + f(x) =∫ 1

0

(Df)(x+ tk + h)kdt

=∫ 1

0

∫ 1

0

(D(Df)(x+ sk + th)(h))kdsdt.

9

Since the left-hand sides of the last two expressions are equal, so are the right-hand sides, and hence∫ 1

0

∫ 1

0

[(D(Df)(x+ th+ sk)(k))h− (D(Df)(x+ th+ sk)(h))k]dsdt = 0.

Since D(Df) is a continuous function, this can only happen if D2f(x)(k, h) =D2f(x)(h, k), for all x, h and k, which is exactly what we needed to prove.

More generally, if f : U → F is a Ck-map, then

Dkf(x0) = D(Dk−1f)(x0) ∈ L(E,Lk−1(E,F )) = Lk(E,F ),

and by an induction based on the previous lemma, we see that in fact

Dkf(x0) ∈ Lks(E,F ) = T ∈ Lk(E,F ) : T is symmetric.

By symmetric we mean that

T (hσ(1), . . . , hσ(k)) = T (h1, . . . , hk),

for all permutations σ in the symmetric group Sk on k letters.It is often useful to have an explicit formula for the higher derivatives of

a composition. The following such formula comes from §3 of [1]. If p and kare positive integers with k ≤ p and (i1, . . . , ik) is a k-tuple of positive integerssuch that i1 + · · ·+ ik = p and i1 ≤ i2 ≤ · · · ik, we define integers σpk(i1, . . . , ik)recursively by requiring that σ1

1(1) = 1 and

σpk(i1, . . . , ik) = δ1i1σ

p−1k−1(i2, . . . , ik) +

k∑j=1

σp−1k (i1, . . . , ij + 1, . . . ik),

where δ1i1

is 1 if i1 = 1 and 0 otherwise.

Proposition 1.2.5. Suppose that U and V are open subsets of Banach spacesE and F respectively, and that f : U → V and g : V → G are Cp maps, whereG is a third Banach space. Then

Dp(g f) =p∑k=1

(Dkg f)P pk (f),

whereP pk (f) =

∑σpk(i1, . . . , ik)(Di1f, . . . ,Dikf),

the sum being taken over all p-tuples of positive integers such that i1+· · ·+ik = pand i1 ≤ i2 ≤ · · · ≤ ik.

The proof of Proposition 1.2.5 is by induction on p starting with the case p = 1,which is an immediate application of the chain rule. For p ≥ 2, one obtains the

10

formula for σpk and the expression for Dp(g f) by applying the chain rule andthe Leibniz rule for differentiating a product.

In order to put Taylor’s theorem in the Banach space setting, we need todefine the integral of a continuous map γ : [0, 1] → E into a Banach space E.The definition is particularly easy if E is a reflexive Banach space; in this case,we just set ∫ 1

0

γ(t)dt = e, where λ(e) =∫ 1

0

λ γ(t)dt.

A definition of the integral for general Banach spaces can be found in [43].Let U be an open subset of a Banach space E. Following [2], we define a

thickening of U to be an open subset U ⊂ E × E such that

1. U × 0 ⊂ U .

2. (x, h) ∈ U ⇒ x+ th ∈ U , for t ∈ [0, 1].

3. (x, h) ∈ U ⇒ x ∈ U .

Proposition 1.2.6. (Taylor’s Theorem.) If a map f : U → F is of class Cr,there exist continuous maps

φk : U → Lks(E,F ), for 1 ≤ k ≤ r and R : U → Lrs(E,F ),

where U is a thickening of U , such that R(x, 0) = 0 and

f(x+ h) = f(x) + φ1(x)h+12!φ2(x)(h, h) + · · ·+ 1

r!φr(x)(h, · · · , h)

+R(x, h)(h, · · · , h).

Here φk(x) = (Dkf)(x), for 1 ≤ k ≤ r.

To see this, we first reduce to the case where F = R by applying the Hahn-Banach Theorem, and then establish via induction,

f(x+ h) = f(x) +∫ 1

0

(Df)(x+ th)hdt

= f(x) + (Df)(x)h+∫ 1

0

[(Df)(x+ th)− (Df)(x)]hdt

= f(x) + (Df)(x)h+∫ 1

0

∫ 1

0

[(D2f)(x+ sth)](h, h)tdsdt

= f(x) + (Df)(x)h+12!

(D2f)(x)(h, h)

+∫ 1

0

∫ 1

0

[(D2f)(x+ sth)−D2f(x)](h, h)tdsdt

11

Continuing in the same manner, we find that

f(x+ h) = f(x) + (Df)(x)h+12!

(D2f)(x)(h, h)

+ · · ·+ 1k!

(Dkf)(x)(h, · · · , h) +R(x, h)(h, · · · , h),

where R(x, h) ∈ Lks(E,F ) depends continuously on x and h and R(x, 0) = 0.

We have seen that many of the basic results of differential calculus of severalvariables extend with little change to the infinite-dimensional context. Thefollowing theorem is somewhat deeper:

Theorem 1.2.7. (Inverse Function Theorem.) If U1 and U2 are opensubsets of Banach spaces E1 and E2 with x0 ∈ U1, and f : U1 → U2 is a C∞ mapsuch that Df(x0) ∈ L(E1, E2) is invertible, then there are open neighborhoodsV1 of x0 and V2 of f(x0), and a C∞ map g : V2 → V1, such that

f g = idV2 and g f = idV1 .

Moreover, Dg(f(x)) = [Df(x)]−1, for x ∈ V1.

Sketch of proof: We can assume without loss of generality that x0 = 0 ∈ U1

and f(0) = 0 ∈ U2. We can assume, moreover, that E1 = E2 and Df(0)is the identity map by replacing f by Df(0)−1 f . Define h : U1 → E1 byh(x) = x − f(x). Then Dh(0) = 0, and by continuity of Dh there exists δ > 0such that

‖x‖ ≤ δ ⇒ x ∈ U1 and ‖dh(x)‖ < 12.

If ‖x‖, ‖y‖ < δ, then it follows from the chain rule that

‖h(x)− h(y)‖ =∣∣∣∣∫ 1

0

d

dt(h(tx+ (1− t)y))dt

∣∣∣∣=∣∣∣∣∫ 1

0

Dh(tx+ (1− t)y)(x− y)dt∣∣∣∣

≤[∫ 1

0

‖Dh(tx+ (1− t)y)‖dt]‖x− y‖ < 1

2‖x− y‖.

More generally, if ‖y‖ < δ/2, and we define the map hy by hy(x) = h(x) + y,then

‖x‖ ≤ δ ⇒ ‖hy(x)‖ ≤ ‖y‖+ ‖h(x)‖ < δ

2+

12‖x‖ ≤ δ,

so hy takes the closed ball of radius δ to itself and

‖hy(x)− hy(x′)‖ = ‖h(x)− h(x′)‖ < 12‖x− x′‖,

12

so hy is a contraction. Thus by the well-known Contraction Lemma, given ywith ‖y‖ < δ/2, there is a unique fixed point x of hy; that is, there is a uniquex such that ‖x‖ ≤ δ and

hy(x) = x ⇒ x− f(x) + y = x ⇒ f(p) = q.

Let

V2 = y ∈ E1 : ‖y‖ < δ/2 and V1 = x ∈ E1 : ‖x‖ < δ, f(x) ∈ V2

and define

g : V2 → V1 by g(y) = x ∈ V1 ⇔ f(x) = y.

Then g is a set-theoretic inverse to f : V1 → V2.To show that g is continuous, it suffices to show that |x−x′| ≤ 2|f(x)−f(x′)|.

But

|x− x′| ≤ |(x− f(x))− (x′ − f(x′))|+ |f(x)− f(x′)|

≤ |h(x)− h(x′)|+ |f(x)− f(x′)| ≤ 12|x− x′|+ |f(x)− f(x′)|,

which clearly implies the desired result.To see that that g is C1, we note first that if x0 ∈ V1,

f(x)− f(x0) = Df(x0)(x− x0) + o(|x− x0|).

If y0 = f(x0) and y = f(x), we can rewrite this equation as

y − y0 = Df(x0)(g(y)− g(y0)) + o(|x− x0|),or g(y)− g(y0) = [Df(x0)]−1(y − y0 + o(|x− x0|)).

The continuity argument shows that [Df(x0)]−1(o(|x− x0|)) is o(|y− y0|), so gis C1 with derivative Dg(y) = (Df)−1(g(y)).

Finally, one uses “bootstrapping”:

g ∈ C1 ⇒ (Df)−1 g ∈ C1 ⇒ Dg ∈ C1 ⇒ g ∈ C2 ⇒ · · ·

to conclude that g is C∞, and the theorem is proven. Later, we will see thatthe technique of bootstrapping used here is a fundamental tool for the study ofnonlinear PDE’s.

Before stating an important corollary of the Inverse Function Theorem, we pointout one of the difficulties in dealing with Banach spaces. A closed linear subspaceE1 of a Banach space E is a Banach space in its own right, but there may notexist a complementary closed subspace E2 such that E is linearly homeomorphicto E1⊕E2. We say that a subspace E1 of a Banach space E is split if there doesexist such a complement E2. For example, any finite-dimensional subspace of a

13

Banach space is split as is any closed subspace of finite codimension. Moreover,any closed subspace of a Hilbert space is split, because the inner product canbe used to define the orthogonal complement.

Corollary 1.2.8. If U is an open subset of the Banach space E with x ∈ U ,and f : U → F is a C∞ map such that Df(x) ∈ L(E,F ) is surjective withsplit kernel, then there exists an open subset V ⊂ U and a diffeomorphismφ : V1×V2 → V , where V1 and V2 are open subsets of Banach spaces E1 and E2

with E = E1 ⊕E2 and E2∼= F , such that f φ is the projection on the second

factor.

Sketch of proof: We can assume without loss of generality that x = 0 and f(0) =0. Let E1 be the kernel of Df(0) and since it splits, let E2 be a complementsuch that E = E1 ⊕ E2. Note that Df(p) establishes an isomorphism from E2

to F . Let

g : E = E1 ⊕ E2 → E = E1 ⊕ E2 by g(x1, x2) = (x1, f(x1, x2)).

One easily checks that Dg(0) is invertible. Now apply the inverse functiontheorem to construct a smooth map

φ : V1 × V2 → V ⊂ U such that g φ = id,

the identity map. The projection on E2 is just f φ.

Remark. Notably absent from our examples of Banach spaces is the space

C∞([0, 1],R) = functions f : [0, 1]→ R whichhave continuous derivatives of all orders ,

which one suspects should have importance for the theory of nonlinear partialdifferential equations. Unfortunately, the natural topology to use on this spaceis not defined by a single norm or seminorm, but by a countable collection ofnorms ‖ · ‖k : k ∈ Z, k ≥ 0, where ‖ · ‖k is defined by (1.2).

Definition. A Frechet space is a vector space E together with a countablecollection of seminorms ‖ · ‖k : k ∈ Z, k ≥ 0 satisfying the following axioms:

1. ‖x‖k = 0 for all k only if x = 0.

2. Suppose that xi is a sequence of elements from E. If for each ε > 0,there is a N such that i, j ≥ N implies that ‖xi − xj‖k < ε for all k, thenthere is an element x ∈ E such that ‖xi − x‖k converges to zero for all k.

Of course every Banach space is a Frechet space, but C∞([0, 1],R) with thecollection of norms defined above is a Frechet space which is not a Banachspace. Given a Frechet space E with seminorms ‖ · ‖k : k ∈ Z, k ≥ 0, we candefine a distance function

d : E × E → R by d(x, y) =∞∑k=0

12k‖x− y‖k,

14

which makes E into a metric space. Thus we can talk of continuous mapsf : E1 → E2 from the Frechet space E1 to the Frechet space E2, and we have acategory consisting of Frechet spaces and continuous linear maps.

Definition. Suppose that E1 and E2 are Frechet spaces, and that U is anopen subset of E1. A continuous map f : U → E2 is said to be continuouslydifferentiable on U if there exists a continuous map Df : U × E1 → E2 suchthat

Df(x)y = limt→0

f(x+ ty)− f(x)t

,

where t ranges throughout R − 0, it being understood that the limit on theright-hand side exists for all x ∈ U and all y ∈ E1.

It is proven in Hamilton’s survey article [32], Part I, 3.2.3 and 3.2.5 that iff : U → E2 is continuously differentiable, the map y 7→ Df(x)y is linear. Thusif E1 and E2 are Banach spaces the above definition agrees with the definitionpreviously given.

We could develop much of the infinite-dimensional calculus and the theoryof infinite-dimensional manifolds in the category of Frechet spaces, and in factthis is carried out in [32], but the Inverse Function Theorem does not hold forFrechet spaces. Moreover, in the existence theory for solutions to nonlinearpartial differential equations, it is often convenient to first prove existence ina given Banach space and then prove regularity using bootstrapping, just aswe did in the proof of the Inverse Function Theorem. This technique seemsparticularly well-adapted to Banach spaces. For these reasons, we prefer tothink of C∞([0, 1],R) as the intersection of a “chain” of Banach spaces,

· · · ⊆ Ck+1([0, 1],R) ⊆ Ck([0, 1],R) ⊆ Ck−1([0, 1],R) ⊆ · · · ⊆ C0([0, 1],R).

For proving theorems, we will usually work in the category of Banach spacesso that we can use theorems like the Inverse Function Theorem. However,the statements of theorems are sometimes more elegant when phrased in thecategory of Frechet spaces.

1.3 Infinite-dimensional manifolds

Definition. Let E be a Banach space. A connected smooth manifold mod-eled on E is a connected Hausdorff space M together with a collection A =(Uα, φα) : α ∈ A, where each Uα is an open subset of M and each

φα : Uα −→ φα(Uα) ⊂ E

is a homeomorphism such that

1.⋃Uα : α ∈ A =M.

2. φβ φ−1α : φα(Uα ∩ Uβ)→ φβ(Uα ∩ Uβ) is C∞, for all α, β ∈ A.

15

A nonconnected Hausdorff space is called a smooth manifold or a Banach mani-fold if each component is a connected smooth manifold modeled on some Banachspace. (There is no harm in allowing different components to be modeled ondifferent Banach spaces.) The smooth manifold is called a Hilbert manifold ifeach component is modeled on a Hilbert space.

We say that A = (Uα, φα) : α ∈ A is the atlas defining the smooth structureon M, and each (Uα, φα) is one of the charts in the atlas.

Remark. We could define Frechet manifolds by simply replacing the phrase“Banach space” by “Frechet space” in the above definition.

Let M and N be smooth manifolds modeled on Banach spaces E and F re-spectively. Suppose that M and N have atlases A = (Uα, φα) : α ∈ A andB = (Vβ , ψβ) : β ∈ B. Then a continuous map F : M → N is said to besmooth if ψβ F φ−1

α is C∞, where defined, for α ∈ A and β ∈ B. It followsfrom the chain rule that the composition of smooth maps is smooth. In this waywe obtain a category whose objects are smooth manifolds modeled on Banachspaces and whose morphisms are smooth maps between such manifolds.

As in the case of finite-dimensional manifolds, a diffeomorphism is a smoothmap between manifolds with smooth inverse. We will often identify two smoothmanifolds if there is a diffeomorphism from one to the other. Later we willconstruct invariants (such as de Rham cohomology) that will often enable us todetermine that two infinite-dimensional manifolds cannot be diffeomorphic.

Of course, the simplest example of a smooth manifold modeled on E is anopen subset U of E in which the atlas is (U, idU ). However, the examples ofmost interest to us will be function spaces.

Example. Suppose that Mn is a complete Riemannian manifold of finite di-mension n, which we can assume is isometrically imbedded in Euclidean spaceRN by the celebrated Nash imbedding theorem [57]. Suppose that Σ is a com-pact smooth manifold of finite dimension m. Then

C0(Σ,RN ) = continuous maps f : Σ→ RN

is a Banach space, and we claim that the subspace

C0(Σ,M) = continuous maps f : Σ→M ⊆ C0(Σ,RN )

is an infinite-dimensional Banach manifold.To construct the charts on C0(Σ,M) we use the exponential map of M .

Suppose that f : Σ → M is a smooth (C∞) map, and consider the Banachspace

C0(f∗TM) = continuous sections of f∗TM,with the C0-norm

‖X‖ = sup|X(p)| : p ∈ Σ,where the | · | on the right is length as defined by the Riemannian metric on M .For ε > 0, we set

Vf,ε = X ∈ C0(f∗TM) : ‖X‖ < ε,

16

and we define

Uf,ε = g ∈ C0(Σ,M) : dM (g(p), f(p)) < ε for all p ∈ Σ,

where dM : M ×M → R is the distance function on M defined by the Rieman-nian metric. Then we define ψf,ε : Vf,ε → Uf,ε by

ψf,ε(X)(p) = expf(p)(X(p)).

By the the proof of the standard tubular neighborhood theorem from finite-dimensional Riemannian geometry, one concludes that when ε > 0 is sufficientlysmall, ψf,ε is a bijection, and we set φf,ε = ψ−1

f,ε . We set

A = (Uf,ε, φf,ε) : f : Σ→M is a C∞ map and ε > 0 issmall enough that ψf,ε is a homeomorphism .

Since smooth maps f : Σ→M are dense in C0(Σ,M), the union of the elementsof A cover C0(Σ,M).

Hence to verify that C0(Σ,M) is a smooth manifold, we need only checkthat φf2,ε2 (φf1,ε1)−1 is smooth where defined, when

(Uf1,ε1 , φf1,ε1), (Uf2,ε2 , φf2,ε2) ∈ A.

Let U be the open subset of the total space of f∗1TM on which (expf2(p))−1 expf1(p) is defined. We can assume that U ∩ Tf1(p)M is convex with compactclosure for each p ∈ Σ, and define

g : U → (total space of f∗2TM)

byg(p, v) = (p, (expf2(p))

−1 expf1(p)(v)), for p ∈ Σ, v ∈ Tf1(p)M .

We can think of g as a family of smooth maps

p 7→ gp : (Tf1(p)M ∩ U)→ Tf2(p)M,

and since Σ is compact and U has compact closure in each fiber, all derivativesDk(gp) are bounded. Using the open neighborhood U of the zero-section, wedefine

U = X ∈ C0(f∗1TM) : X(Σ) ⊆ U

and define ωg : U → C0(f∗2TM) by ωg(X) = g X.To finish the proof that C0(Σ,M) is a smooth manifold, it suffices to show

that ωg is smooth. We can formalize this statement in a theorem, known as theω-lemma:

Lemma 1.3.1. Suppose that E1 and E2 are finite-dimensional vector bundlesover the compact smooth manifold Σ and that U is a bounded open neighbor-hood of the zero section of E1 whose restriction to each fiber of E1 is convex.

17

If g : U → (total space of E2) is a smooth map which takes the fiber of E1 overp to the fiber of E2 over p (for each p ∈ Σ), and

U = σ ∈ C0(E1) : σ(Σ) ⊆ U,

then the map ωg : U → C0(E2), defined by ωg(σ) = g(σ), is smooth.

The first step is to show that ωg is continuous; this is straightforward and wecan safely leave it to the reader.

Suppose now that σ, η, σ + η ∈ U . It follows from Taylor’s theorem that foreach p ∈ Σ,

gp((σ + η)(p)) = gp(σ(p)) +Dgp(σ(p))η(p) +R(σ, η)(p)(η(p)),

where

R(σ, η)(p)(η(p)) =[∫ 1

0

[Dgp((σ + tη)(p))−Dgp(σ(p))]dt]η(p). (1.3)

We can write this as

ωg(σ + η) = ωg(σ) + T (σ)(η) +R(σ, η)η,

whereT (σ)(η)(p) = Dgp(σ(p))η(p),

and R(σ, η)η is the remainder term given by (1.3). Note that

‖T (σ)(η)‖ = supp∈Σ|Dgp(σ(p))η(p)| ≤ sup

p∈Σ|Dgp(σ(p))|‖η‖,

so T (σ) is a continuous linear map from a neighborhood of 0 in C0(E1) toC0(E2). Thus T extends to a linear map from C0(E1) to C0(E2). We nextestimate the error term R(σ, η)η; since∣∣∣∣∫ 1

0

[Dgp((σ + tη)(p))−Dgp(σ(p))]dt∣∣∣∣ ≤ |D2gp(σ(p))||η(p)|,

we conclude that

‖R(σ, η)η‖ ≤ supp∈Σ|D2gp(σ(p))‖η‖2 = o(η).

This implies that ωg has the derivative Dωg(σ) = T (σ) at σ. We have defineda map

Dωg : U −→ L(C0(E1), C0(E2)), (1.4)

and it is relatively straightforward to show that Dωg is continuous, showingthat ωg is C1.

18

We would like to extend this argument to higher derivatives, and for this weneed to factor the derivative given by (1.4) as follows: Recalling that g : U →(total space of E2), we define a corresponding map

Dg : U → (total space of L(E1, E2)) by setting Dg(p) = Dgp.

(We can regard Dg as a “partial derivative” of g in which the point p ∈ Σ isheld fixed.) We can then define

ωfDg : U → C0(Σ, L(E1, E2)) by ωfDg(σ)(p) = Dgp(σ(p)).

Using the fact that C0(Σ,R) is a Banach algebra, we can show that the map

A : C0(Σ, L(E1, E2))→ L(C0(Σ, E1), C0(Σ, E2))defined by A(T )(σ)(p) = T (p) · σ(p),

is smooth, thus providing the desired factorization,

Dωg = A · ωfDg.The argument presented in the preceding paragraph now shows that ωfDg is C1,from which we conclude that Dωg is C1, and hence ωg is C2.

Next, we show that ωgD2gis C1, which implies that ωg is C3, and so forth.

By induction, we establish that ωg is Ck for all k ∈ N, and hence ωg is C∞.

The above lemma has the following consequence:

Theorem 1.3.2. If Σ and M are smooth manifolds with Σ compact, thenC0(Σ,M) is a smooth manifold modeled on the Banach spaces

C0(f∗TM) = continuous sections of f∗TM,

where f : Σ → M is a smooth map. Moreover, if g : M → N is a C∞ map,then the map

ωg : C0(Σ,M)→ C0(Σ, N) defined by ωg(f) = g f,

is smooth.

To prove the moreover part of the theorem, we need to show that if f1 : Σ→Mand f2 : Σ→ N are smooth, then

φf2,ε2 ωg φ−1f1,ε1

is smooth where defined.

The proof of this is a straightforward application of Lemma 1.2.1.

A modification of the previous example is often quite useful. Let Σ be a compactsmooth manifold of finite dimension m with boundary ∂Σ, Mn a complete

19

Riemannian manifold of finite dimension n and f0 : ∂Σ → M a fixed smoothmapping. We claim that

C00 (Σ,M) = continuous maps f : Σ→M : f |∂Σ = f0.

is a Banach manifold. The component containing f is modeled on the Banachspace

C00 (f∗TM) = continuous sections of f∗TM

which vanish on the boundary of M.

The proof is a straightforward modification of the proof of Theorem 1.2.2.Unfortunately, the manifolds C0(Σ,M) are not sufficient for constructing a

global theory of partial differential equations. We need to be able to differenti-ate elements in our function spaces. Thus we need to start off with a somewhatstronger Banach space than the space C0(Σ,R) of continuous real-valued func-tions on Σ.

Thus for k ∈ N, we are led to consider the space Ck(Σ,R) of real-valuedfunctions on Σ which have continuous derivatives up to order k, a Banach spacewith respect to the norm

‖f‖Ck = sup‖f‖(p) + ‖Df‖(p) + · · ·+ ‖Dkf‖(p) : p ∈ Σ. (1.5)

In fact, it is easily checked that if f, g ∈ Ck(Σ,R), then

‖fg‖Ck ≤ ‖f‖Ck‖g‖Ck , (1.6)

so Ck(Σ,R) is in fact a Banach algebra.More generally, we can consider the space Ck(Σ,RN ) of RN -valued functions

on Σ which have continuous derivatives up to order k, which is also a Banachspace with norm defined by (1.5). The Banach algebra condition (1.6) ensuresthat we can define a continuous multiplication

µ : Ck(Σ, L(RN ,RM ))× Ck(Σ,RN ) −→ Ck(Σ,RM )

by simply multiplying in the range.If M is an n-dimensional Riemannian manifold which isometrically imbedded

in Rn, we let

Ck(Σ,M) = f ∈ Ck(Σ,RN ) : f(p) ∈M for all p ∈ Σ .

We claim that Ck(Σ,M) is a smooth manifold.The construction of the atlas is just like the construction for C0(Σ,M). For

ε > 0, we setVf,ε = X ∈ Ck(f∗TM) : ‖X‖C0 < ε,

an open subset of Ck(f∗TM), and we define

Uf,ε = g ∈ C0(Σ,M) : dM (g(p), f(p)) < ε for all p ∈ Σ.

20

As before, we define ψf,ε : Vf,ε → Uf,ε by

ψf,ε(X)(p) = expf(p)(X(p)).

When ε > 0 is sufficiently small, ψf,ε is a bijection, and we set φf,ε = ψ−1f,ε . As

smooth atlas for Ck(Σ,M), we take

A = (Uf,ε, φf,ε) : f : Σ→M is a C∞ map and ε > 0 issmall enough that ψf,ε is a homeomorphism .

Just as before, since smooth maps f : Σ → M are dense in Ck(Σ,M), theunion of the elements of A cover Ck(Σ,M). To finish the proof that Ck(Σ,M)is an infinite-dimensional smooth manifold, we need to verify once again thatφf2,ε2 (φf1,ε1)−1 is smooth where defined, when

(Uf1,ε1 , φf1,ε1), (Uf2,ε2 , φf2,ε2) ∈ A.

But this follows from a corresponding ω-lemma:

Lemma 1.3.3. Suppose that E1 and E2 are finite-dimensional vector bundlesover the compact smooth manifold Σ and that U is a bounded open neighbor-hood of the zero section of E1 whose restriction to each fiber of E1 is convex.If g : U → (total space of E2) is a smooth map which takes the fiber of E1 overp to the fiber of E2 over p (for each p ∈ Σ), and

U = σ ∈ Ck(E1) : σ(Σ) ⊆ U,

then the map ωg : U → Ck(E2), defined by ωg(σ) = g(σ), is smooth.

The proof is virtually identical to the proof given for Lemma 1.2.1. The proofextends to Ck maps because the space Ck(Σ,R) has two key properties:

1. It is a Banach algebra, and

2. there is be a continuous inclusion from Ck(Σ,R) into the Banach algebraC0(Σ,R) of continuous functions.

Thus just as before, we can construct an important family of Banach manifolds:

Theorem 1.3.4. If Σ and M are smooth manifolds with Σ compact, then foreach k ∈ N, Ck(Σ,M) is a smooth manifold modeled on the Banach spaces

Ck(f∗TM) = Ck sections of f∗TM,

whenever f : Σ→M is a smooth map. Moreover, if g : M → N is a C∞ map,then the map

ωg : Ck(Σ,M)→ Ck(Σ, N) defined by ωg(f) = g f,

is smooth.

21

To summarize, for each pair (Σ,M) of finite-dimensional smooth manifolds,with Σ compact, we have a chain of Banach manifolds,

· · · ⊆ Ck+1(Σ,M) ⊆ Ck(Σ,M) ⊆ Ck−1(Σ,M) ⊆ · · · ⊆ C0(Σ,M).

The intersection of these manifolds is the space C∞(Σ,M) of C∞ maps from Σto M , which could be made into a Frechet manifold, but we will not enter intothe details of that here.

We can now try to formulate calculus of variations problems in terms of theinfinite-dimensional manifolds that we have constructed. Thus, for example, wecan define the action function

J : C1(S1,M) −→ R by J(γ) =12

∫S1|γ′(t)|2dt,

and check without much difficulty that J is a smooth map. As we learnedin elementary differential geometry courses, the “critical points” for J are thesmooth closed geodesics on M .

Suppose that Σ is a compact two-dimensional Riemann surface. Thus wecan imagine that Σ has a Riemannian metric, but we forget about the metricexcept for its conformal equivalence class, which we denote by ω. Suppose that

(Uα, (xα, yα)) : α ∈ A

is an atlas of isothermal coordinate charts on Σ, and let ψα : α ∈ A bea partition of unity subordinate to Uα : α ∈ A. We can then define theDirichlet integral

Eω : C1(S1,M) −→ R by Eω(f) =12

∫Σ

∑α∈A

ψα

[∣∣∣∣ ∂f∂xα∣∣∣∣2 +

∣∣∣∣ ∂f∂yα∣∣∣∣2]dxαdyα.

Once again it is relatively easy to check that Eω is a smooth map on the infinite-dimensional manifold C1(Σ,M). Later we will see that the “critical points” forEω are harmonic maps.

More generally, suppose that Σ is an m-dimensional Riemannian manifoldwith Riemannian metric expressed in local coordinates (1, . . . xm) on Σ by

h =m∑

a,b=1

ηabdxadxb.

If f : Σ → M ⊆ RN is a smooth map and (ηab) denotes the matrix inverse to(ηab), we set

|df |2 =m∑

a,b=1

ηab∂X

∂xa· ∂X∂xb

and dA =√

det(ηab)dx1 · · · dxm.

We can then define the Dirichlet integral

E : C1(Σ,M) −→ R by E(f) =12

∫Σ

|df |2dA,

22

which is once again a smooth real-valued function on the infinite-dimensionalmanifold C1(Σ,M). In the case where the domain is two-dimensional, choiceof isothermal parameters leads to exactly the same integrand as before, so thisgeneralizes the previous energy functions to higher dimensional domains.

1.4 The basic mapping spaces

For the study of geodesics, harmonic and minimal surfaces and pseudoholomor-phic curves, as well as other nonlinear partial differential equations, we need acollection of function spaces with weak enough topologies that it is relativelyeasy to prove convergence of a sequence which is tending toward an infimumof energy on a given component. The infinite-dimensional manifolds that haveproven to be most useful in this regard are those modeled on Sobolev spaces.In this section, we describe the simplest of these spaces.

If Σ is a compact Riemannian manifold, we can define an inner product (·, ·)on the space C∞(Σ,R)) of smooth real-valued maps by

(f, g) =∫

Σ

(fg + 〈Df,Dg〉)dA,

where the inner product 〈·, ·〉 on the right is the usual inner product in thecotangent space defined by the Riemannian metric on Σ, and dA denotes thearea or volume form on Σ. The inner product (·, ·) makes space C∞(Σ,R))of smooth functions into a pre-Hilbert space. Any pre-Hilbert space has aHilbert space completion, the set of equivalence classes of Cauchy sequences, asdescribed at the beginning of §1.2. The Hilbert space completion in our case isdenoted by L2

1(Σ,R)), and is called the Sobolev space of L21-functions on Σ.

A second important Sobolev space generalizes the Lp spaces studied in realanalysis when 1 < p <∞. We start by defining a norm ‖ · ‖Lp1 on C∞(Σ,R)) by(

‖f‖Lp1)p

=∫

Σ

(|f |p + |Df |p)dA,

where |Df | is the length calculated with respect to the Riemannian metricon Σ. This norm makes C∞(Σ,R)) into a pre-Banach space. As before, wecan construct the Banach space completion. This Banach space completion isdenoted by Lp1(Σ,R)) and is called the Sobolev space of Lp1-functions on Σ. Ofcourse, when p = 2 this reduces to the previous example.

We can also define versions of these Sobolev spaces for higher numbers ofderivatives. Thus we can define a norm ‖ · ‖Lpk on C∞(Σ,R)) by(

‖f‖Lpk)p

=∫

Σ

(|f |p + |Df |p + · · ·+ |Dfk|p)dA,

and construct the completion with respect to this norm, which is denotedLpk(Σ,R). The resulting space is a Banach space, and a Hilbert space whenp = 2. We thus obtain a chain of Banach spaces,

· · · ⊆ Lpk(Σ,R) ⊂ · · · ⊆ Lp1(Σ,R) ⊆ Lp(Σ,R),

23

and the intersection of all the spaces in the chain is just the space C∞(Σ,R)) ofsmooth functions. These spaces are essential for the modern theory of partialdifferential equations and they are compared by means of the Sobolev Lemma,which in general form states

Lpk(Σ,R) ⊆ Cl(Σ,R) for p(k − l) > dim(Σ), (1.7)

where ⊆ indicates continuous inclusion. A complement to the Sobolev Lemmastates that for pk > dim(Σ), Lpk(Σ,R) is a Banach algebra in an appropri-ate norm defining the “Banachable” structure. Additional information on theSobolev spaces is found in standard references, such as Evans [19].

For the case where Σ = S1, where S1 is the unit interval [0, 1] with endpointsidentified, the Sobolev Lemma is relatively easy to establish, and we do thathere:

Lemma 1.4.1. There is a continuous linear injection i : L21(S1,R)→ C0(S1,R)

which extends the inclusion C∞(S1,R) ⊂ C0(S1,R).

Proof: We begin with the sequence of inequalities:

|f(t)| ≤ |f(τ)|+∫ τ

t

|f ′(u)|du ≤ |f(τ)|+∫S1|f ′(u)|du.

Averaging over τ and using the Cauchy-Schwarz inequality yields

|f(t)| ≤[∫

S1|f(u)|du+

∫S1|f ′(u)|du

]≤[∫

S1[|f(u)|2 + |f ′(u)|2]du

]1/2

= (f, f)1/2,

where (·, ·) denotes the L21 inner product. Taking the supremum over all t yields

‖f‖C0 ≤ ‖f‖L21.

Thus a Cauchy sequence with respect to the L21 inner product gets taken under

the inclusion C∞(S1,R) ⊂ C0(S1,R) to a Cauchy sequence with respect to theC0-norm. By definition, an element of L2

1 is an equivalence class of Cauchysequences, and the map i is defined by sending this equivalence class to thelimit of the C0 Cauchy sequence. It is immediate that i is injective.

Lemma 1.4.2. L21(S1,R) is a Banach algebra; multiplication of functions is a

continuous bilinear map

L21(S1,R)× L2

1(S1,R) −→ L21(S1,R).

24

Proof: It follows from the Cauchy-Schwarz inequality that

‖fg‖2L21

= (fg, fg) =∫S1

[(fg)2 + [(fg)′]2]dt =∫S1

[(fg)2 + (f ′g + fg′)2]dt

=∫S1

[(fg)2 + (f ′)2g2 + 2fgf ′g′ + f2(g′)2]dt

≤∫S1

[(fg)2 + (f ′)2g2 + f2(g′)2

]dt+ 2‖fg‖C0

∫S1

(f ′g′)dt

≤ ‖f‖2C0‖g‖2L21

+ ‖g‖2C0‖f‖2L21

+ ‖f‖2C0‖g‖2L21

+ 2‖fg‖C0‖f‖L21‖g‖L2

1.

Since the C0 norm is less than the L21, we find that

‖fg‖2L21≤ ‖f‖2L2

1‖g‖2L2

1,

finishing the proof of the Lemma.

It follows from this Lemma that the multiplication map

L21(S1,Hom(Rm,Rn))× L2

1(S1,Rm) −→ L21(S1,Rn)

is continuous.Suppose now that M is a complete connected finite-dimensional Rieman-

nian manifold isometrically imbedded as a proper submanifold of an ambientEuclidean space RN . We let

L21(S1,M) = γ ∈ L2

1(S1,RN ) : γ(t) ∈M for all t ∈ S1,

which is a closed subspace of L21(S1,RN ) by Lemma 1.4.1.

We claim that L21(S1,M) is an infinite-dimensional smooth manifold, the

proof being just like the proof for C0(S1,M). If γ : S1 →M is a smooth curve,we let

Vγ,ε = X ∈ L21(S1, γ∗TM) : the L2

1 norm of X is < ε,Uγ,ε = λ ∈ L2

1(S1,M) : dM (λ(t), γ(t)) < ε for all t ∈ S1,

and if ε > 0 is sufficiently small, we define

ψγ,ε : Vγ,ε → Uγ,ε by ψγ,ε(X)(p) = expγ(t)(X(t)), φγ,ε = ψ−1γ,ε.

We then set

A =

(Uγ,ε, φγ,ε) : γ : S1 →M is a C∞ map and ε > 0 issmall enough that ψγ,ε is a homeomorphism .

and prove that it is a smooth atlas by the same argument used to establishLemma 1.3.1. Thus we obtain:

25

Theorem 1.4.3. If M is a smooth manifold, then L21(S1,M) is a smooth

manifold modeled on the Banach spaces L21(γ∗TM) for γ : S1 → M a smooth

map. Moreover, if g : M → N is a C∞ map, then the map

ωg : L21(S1,M)→ L2

1(S1, N) defined by ωg(γ) = g γ,

is also C∞.

We are also interested in Lp1-maps from a compact oriented surface Σ. It turnsout that these are Holder continuous in accordance with the following definition.

Definition. If Σ is a metric space, a map f : Σ → R is said to be Holdercontinuous with Holder exponent γ ∈ (0, 1] if there is a constant C > 0 suchthat

|f(p)− f(q)| ≤ Cd(p, q)γ for all p, q ∈ Σ.

We let C0,γ(Σ,R) be the space of all functions f : Σ → R which are Holdercontinuous. If f ∈ C0,γ(Σ,R), we let

[f ]C0,γ = supf(p)− f(q)

(d(p, q)γ: p, q ∈ Σ, p 6= q

.

Lemma 1.4.4. If Σ is a compact oriented surface and p > 2, there is a con-tinuous linear injection i : Lp1(Σ,R) → C0,γ(Σ,R), where γ = 1 − 2/p, whichextends the inclusion C∞(Σ,R) ⊂ C0,γ(Σ,R). Moreover, there is a constantC > 0 such that

[f ]C0,γ ≤ C‖f‖Lp1 .

A complete proof of this is given in Evans [19], §5.6.2. We only prove theweaker result that Lp1(Σ,R) ⊆ C0,γ(Σ,R). We begin by assuming that Σ isthe torus with flat Riemannian metric expressed in terms of suitable conformalcoordinates as ds2 = dx2 + dy2. We consider a smooth function f(x, y) on thedisk D(p, r0) of radius r0 about p ∈ Σ defined in terms of Euclidean coordinatescentered at p by x2 + y2 ≤ r2

0, then after shifting to polar coordinates (r, θ)defined by

x = r cos θ, y = r sin θ,

we see that

|f(s, θ)− f(p)| =∫ s

0

∂f

∂r(r, θ)dr ≤

∫ s

0

|Df |(r, θ)dr,

and hence∫ 2π

0

|f(s, θ)− f(p)|dθ ≤∫ 2π

0

∫ s

0

|Df |(r, θ)drdθ ≤∫D(p,r0)

|Df |r

dxdy.

Thus ∫ 2π

0

∫ r0

0

|f(s, θ)− f(p)|sdsdθ ≤[∫ r0

0

rdr

][∫D(p,r0)

|Df |r

dxdy

]

26

and hence ∫D(p,r0)

|f(x, y)− f(p)|dxdy ≤ r20

2

∫D(p,r0)

|Df |r

dxdy.

It follows from the Holder inequality that

r20

2

∫D(p,r0)

|Df |r

dxdy ≤ r20

2

[∫D(p,r0)

|Df |pdxdy

]1/p [∫D(p,r0)

dxdy

rp/(p−1)

](p−1)/p

while direct integration yields∫D(p,r0)

dxdy

rp/(p−1)=∫D(p,r0)

rp/(1−p)dxdy

=∫ 2π

0

∫ r0

0

r1/(1−p)drdθ =2π(p− 1)p− 2

r(p−2)/(p−1)0 .

Thus∫D(p,r0)

|f(x, y)− f(p)|dxdy ≤ r20

2

(2π(p− 1)p− 2

)(p−1)/p

r(p−2)/p0 ‖Df‖Lp . (1.8)

It follows from (1.8) and the Holder inequality that

πr20|f(0)| ≤

∫D(p,r0)

|f(x, y)− f(p)|dxdy +∫D(p,r0)

|f(x, y)|dxdy

≤ (constant)‖Df‖Lp + (constant)‖f‖L1

≤ (constant)‖Df‖Lp + (constant)‖f‖Lp(area of D(p, r0))(p−1)/p

≤ (constant)‖f‖Lp1 ,

which quickly yields the desired result when Σ is the flat torus.If Σ is a more general Riemann surface, we can give Σ a Riemannian metric of

constant curvature of constant curvature and total volume one. Choose r0 > 0less than the injectivity radius of this metric. A modification of the aboveargument can then be applied to a normal coordinate ball of radius r0 showingthat if p ∈ Σ, then

|f(p)| ≤ (constant)‖f‖Lp1 , and hence ‖f‖C0 ≤ (constant)‖f‖Lp1 .

(Note that changing the Riemannian metric on Σ merely replaces the Lp1-normby an equivalent norm, so adopting the constant curvature metric imposes norestriction.) Thus if a sequence fi of smooth functions on Σ converges to alimit in Lp1-norm, fi converges also in C0 norm to a unique limit functionf∞ ∈ C0. Thus any element of Lp1(Σ,R) can be identified with a continuousfunction and the Lemma is proven.

27

Remark 1.4.5. The previous Sobolev Lemma for Lp1 maps from a surfacebegins to fail as p approaches 2 from above. The reason is that the highestorder term in the L2

1-norm is conformally invariant and hence invariant underdilations. In the case where D is the unit disk centered at the origin in R2 thishighest order term is ∫

D

|Df |2dxdy.

In contrast, the highest order term in the Lp1-norm is not invariant under dila-tions. Given a smooth map f : D → RN which takes the boundary ∂Dε to apoint, we can define a dilated map fε : Dε → RN , where Dε is the ball of radiusε > 0 centered at the origin in R2, by

fε(x, y) = f(xε,y

ε

). Then |Dfε(x, y)| = 1

ε

∣∣∣Df (xε,y

ε

)∣∣∣ .and if x = x/ε and y = y/ε denote the coordinates corresponding to x and y onthe unit disk D1,∫

Dε

|Dfε|pdxdy =(

1ε

)(p−2) ∫D1

|Df |2αdxdy.

Thus as ε → 0, the Lp1-norm of fε approaches infinity as long as p > 2. Inparticular, a bound on the L2

1-norm does not imply a bound on the C0-norm.

Lemma 1.4.6. If Σ is a compact oriented surface and p > 2, Lp1(Σ,R) satisfies

‖fg‖Lp1 ≤ 2‖f‖Lp1‖g‖Lp1 .

Thus after passing to an equivalent norm, we can show that Lp1(Σ,R) is a Banachalgebra.

Sketch of Proof: By the previous Lemma,

‖fg‖Lp ≤ ‖f‖C0‖g‖Lp + ‖g‖C0‖f‖Lp ≤ ‖f‖Lp1‖g‖Lp + +‖g‖Lp1‖f‖Lp

and

‖D(fg)‖Lp ≤ ‖f‖C0‖Dg‖Lp +‖g‖C0‖Df‖Lp ≤ ‖f‖Lp1‖Dg‖Lp ++‖g‖Lp1‖Df‖Lp .

Adding these two inequalities yields the statement of the Lemma.

If Σ is a compact surface and p > 2, we let

Lp1(Σ,M) = f ∈ Lp1(Σ,RN ) : f(p) ∈M for all p ∈ Σ,

a closed subspace of Lp1(Σ,RN ) by Lemma 1.4.3.If Σ is a compact surface and p > 2, we claim that Lp1(Σ,M) is an infinite-

dimensional smooth manifold. In this case, when f : Σ→M is a smooth curve,we let

Vγ,ε = X ∈ Lp1(Σ∗TM) : the Lp1 norm of X is < ε,Uγ,ε = g ∈ Lp1(Σ,M) : dM (f(p), g(p)) < ε for all p ∈ Σ,

28

and if ε > 0 is sufficiently small, we define

ψγ,ε : Vγ,ε → Uγ,ε by ψγ,ε(X)(p) = expf(p)(X(p)), φγ,ε = ψ−1γ,ε.

We then set

A = (Uγ,ε, φγ,ε) : f : Σ→M is a C∞ map and ε > 0 issmall enough that ψγ,ε is a homeomorphism .

and once again prove that it is a smooth atlas by the same argument used toestablish Lemma 1.3.1. Thus we obtain:

Theorem 1.4.7. If Σ is a compact smooth surface and M is a smooth manifold,then for p > 2, Lp1(Σ,M) is a smooth manifold modeled on the Banach spacesLp1(f∗TM) for f : Σ → M a smooth map. Moreover, if g : M → N is a C∞

map, then the map

ωg : Lp1(Σ,M)→ Lp1(Σ, N) defined by ωg(γ) = g γ,

is also C∞.

In exactly the same way, we could show that if Σ is an m-dimensional smoothmanifold and p > m, then Lp1(Σ,M) is an infinite-dimensional smooth manifold.

1.5 Homotopy type of the space of maps

A continuous map f : X → Y between topological spaces is said to be a homo-topy equivalence if there is a continuous map g : Y → X such that f g andg f are both homotopic to the identity. The following theorem was provenquite early in the theory of manifolds of maps; see Eells [16] for the appropriatereferences.

Theorem 1.5.1. Let M be a compact connected Riemannian manifold. Thenthe inclusions

Ck(S1,M) ⊂ L21(S1,M) and Ck(S1,M) ⊂ C0(S1,M)

are homotopy equivalences when k ≥ 1.

The point of this Theorem is that from the point of view of homotopy theory,L2

1(S1,M) is essentially the same as the space of continuous maps C0(S1,M)with the compact open topology. This latter space has been extensively studiedby topologists and much is known about its homotopy and homology groups, aswe will see later.

The proof of the Theorem is an application of the theory of “smoothing opera-tors.”

29

For preparation, we suppose that φ : R→ R is a smooth map which vanishesoutside [−1, 1]. Suppose, moreover, that

φ ≥ 0 and∫

Rφ = 1.

For ε > 0, let φε(t) = (1/ε)φ(t/ε), so that

supp(φε) ⊂ [−ε, ε] and∫

Rφε = 1.

If γ ∈ C0(S1,RN ), we can regard γ as an element of C0(R,RN ) such thatγ(t+ 1) = γ(t) for all t, and we define φε ∗ γ ∈ C0(R,RN ) by

(φε ∗ γ)(t) =∫

Rφε(t− τ)γ(τ)dτ =

∫Rφε(s)γ(t− s)ds.

It is immediately checked that φε ∗ γ is C∞ and

dk

dtk(φε ∗ γ)(t) =

dk

dtk(φε) ∗ γ =

∫R

(dk

dtkφε

)(t− τ)γ(τ)dτ.

Moreover, (φε ∗ γ)(t+ 1) = (φε ∗ γ)(t), and hence φε ∗ γ ∈ C∞(S1,RN ). We canthus define smoothing operators

Sε : C0(S1,RN )→ Ck(S1,RN ), Sε : L21(S1,RN )→ Ck(S1,RN )

by Sε(γ) = φε ∗ γ. It is not difficult to show that the maps

Sε : C0(S1,RN )→ Ck(S1,RN ), Sε : L21(S1,RN )→ Ck(S1,RN )

are continuous.

Proof of Theorem 1.5.1: Recall that we regard M as a submanifold of RN .Choose δ > 0 so small that the exponential map

exp : NM → RN , defined by exp(v) = p+ v, for v ∈ TpM ,

maps NMδ = v ∈ NM : |v| < δ diffeomorphically onto

M(δ) = p ∈ RN : d(p,M) < δ.

Then the “nearest point projection” map r : M(δ)→M , defined by r(p+v) = pfor p ∈M , is a strong deformation retraction from M(δ) to M . To see this, wedefine

h : M(δ)× [0, 1]→M(δ) by h(p+ v, t) = p+ (1− t)v,

and check that

1. h(q, 0) = q, for q ∈M(δ),

30

2. h(q, 1) = r(q) ∈M , for q ∈M(δ), and

3. h(p, t) = p, for p ∈M .

We have a similar strong deformation retraction on the function space level.The ω-Lemma gives us a smooth map

ωh : L21(S1,M(δ)× [0, 1])→ L2

1(S1,M(δ)) defined by ωh(γ) = h γ.

We define

j : L21(S1,M(δ))× [0, 1]→ L2

1(S1,M(δ)× [0, 1]) by j(γ, τ)(t) = (γ(t), τ)

and let H = ωh j. Then

1. H(γ, 0) = γ, for γ ∈ L21(S1,M(δ)),

2. H(γ, 1) = r γ ∈ L21(S1,M), for γ ∈ L2

1(S1,M(δ)), and

3. H(γ, t) = γ, for γ ∈ L21(S1,M).

We can therefore define a strong deformation retraction

R : L21(S1,M(δ)) −→ L2

1(S1,M)

by R(γ) = ωr(γ) = H(γ, 1). In a similar fashion, we can define a strongdeformation retraction

R : Ck(S1,M(δ)) −→ C0(S1,M),

whenever k ≥ 0.Let εk = 2−k and let

C0(S1,M)εk = γ ∈ C0(S1,M) : γ maps the closed interval

[(m− 1)2−k, (m+ 1)2−k] into the open ball

B(γ(m2−k); δ) for each integer m such that 0 ≤ m ≤ 2k,

an open set in the compact-open topology. They key point of this set is thatwhen |s − t| < 2−k then the straight line from γ(s) to γ(t) in RN lies entirelywithin M(δ) hence Sε ? γ lies within M(δ) when ε ≤ εk. Note that

C0(S1,M) =∞⋃k=1

C0(S1,M)εk , L21(S1,M)εk+1 ⊂ L2

1(S1,M)εk ,

and we therefore say that C0(S1,M) is a monotone union of the subspacesC0(S1,M)εk . Similarly, we let

L21(S1,M)εk = L2

1((S1,M) ∩ C0(S1,M)εk ,

Ck(S1,M)εk = Ck((S1,M) ∩ C0(S1,M)εk , when k ≥ 1,

31

thereby expressing L21(S1,M) and Ck(S1,M) as monotone unions for k ≥ 1.

By analogous formulae, we define

C0(S1,M(δ))εk , L21(S1,M(δ))εk and Ck(S1,M(δ))εk ,

for k ≥ 1. We can then define smoothing operators

Sεk : L21(S1,M)εk −→ Ck(S1,M(δ))εk ,

sinceγ ∈ L2

1(S1,M)εk ⇒ Sεk ? γ ∈ Ck(S1,M(δ))εk .

We define s to the composition of

Sεk : L21(S1,M)εk → Ck(S1,M(δ))εk and

R : Ck(S1,M(δ))εk → Ck(S1,M)εk .

We claim that if i : Ck(S1,M)εk ⊂ L21(S1,M)εk is the inclusion, then

s i and i s

are homotopic to the identity. This is easy to verify. To get the homotopy froms i to the identity, we simply define

H1 : Ck(S1,M)εk × [0, 1]→ Ck(S1,M)εk

by H1(γ, t) = R (tSεk + (1− t)id)) i(γ).

Similarly, to get the homotopy from i s to the identity, we define

H2 : L21(S1,M)εk × [0, 1]→ L2

1(S1,M)εk

by H2(γ, t) = i R (tSεk + (1− t)id)(γ),

This shows that for each k ∈ N, the inclusion

Ck(S1,M)εk ⊂ L21(S1,M)εk

is a homotopy equivalence.To finish the proof, we must take an appropriate limit as k → ∞. Suppose

that the metrizable space X is a monotone union of open subsets, by which wemean that we have a sequence of spaces

U1 ⊂ U2 ⊂ U3 ⊂ · · · such that X =⋃Uk : k ∈ N.

Suppose, moreover, that we let

X? = (U1 × [1, 2]) ∪ (U2 × [2, 3]) ∪ (U3 × [3.4]) ∪ · · ·

topologized as a subset of X × R. We say that X is the homotopy direct limitof the subspaces Uk : k ∈ N if the projection p : X? → X on the first factor

32

is a homotopy equivalence. If the subsets Uk are open, then the open coverUk : k ∈ N has a C0 subordinate partition of unity ψk : k ∈ N. In this case,the map

f : X → X? defined by f(x) =

(x,

∞∑k=1

kψk(x)

)is a homotopy inverse to p, showing that X is a homotopy direct limit in thiscase. Using this argument, one easily verifies that Ck(S1,M) is a homotopydirect limit of its subspaces Ck(S1,M)εk and L2

1(S1,M) is a homotopy directlimit of L2

1(S1,M)εk .Now we apply the following Lemma, which is just Theorem A from the

Appendix to Milnor’s book on Morse theory [50]:

Lemma 1.5.2. Suppose that X is the homotopy direct limit of Uk : k ∈ Nand that Y is the homotopy direct limit of Vk : k ∈ N. If f : X → Y isa continuous map such that f(Uk) ⊆ Vk and the restriction of f to Uk is ahomotopy equivalence from Uk to Vk, then f itself is a homotopy equivalence.

We refer the reader to Milnor for the proof of this Lemma. It implies that theinclusion Ck(S1,M) ⊂ L2

1(S1,M) is a homotopy equivalence when k ≥ 1. Ina similar manner, one verifies that the inclusion Ck(S1,M) ⊂ C0(S1,M) is ahomotopy equivalence when k ≥ 1.

Theorem 1.5.3. Let M be a compact connected Riemannian manifold, Σ acompact connected Riemann surface, p > 2. Then the inclusions

Ck(Σ,M) ⊂ Lp1(Σ,M), Ck(Σ,M) ⊂ C0(Σ,M)

are homotopy equivalences.

The proof is essentially the same as for the previous theorem, with 2 replaced byp, except for the definition of smoothing operators defined on Σ. The construc-tion of such operators is a standard technique in the theory of partial differentialequations. We describe only the case Σ = T 2 here, where T 2 = C/Λ, Λ beinga lattice in C; the construction in this case is particularly transparent. (In thegeneral case, the ideas are the same, but one constructs the smoothing operatorsby piecing together using a partition of unity on Σ.)

Note that an element f ∈ Lp1(T 2,RN ) can be regarded as a map f : C→ RNsuch that f(z + λ) = f(z) for λ ∈ Λ.

Suppose that φ : C → [0,∞) is a smooth map which vanishes outside D =z ∈ C : |z| ≤ 1 such that ∫

Cφ dxdy = 1,

where (x, y) are the standard coordinates on C. Let φε(z) = (1/ε2)φ(z/ε), sothat

supp(φε) ⊂ z ∈ C : |z| ≤ ε and∫

Cφεdxdy = 1.

33

If f : C → M comes from an element f ∈ Lp1(T 2,RN ), we define φε ∗ f ∈C∞(C,RN ) by

(φε ∗ γ)(z) =∫

Cφε(z − w)γ(w)dw.

It is immediately checked that (φε ∗ f)(z+λ) = (φε ∗ f)(z) for λ ∈ Λ, so (φε ∗ f)can be identified with an element of C∞(T 2,RN ).

Thus we can define smoothing operators

Sε : C0(T 2,RN )→ Ck(T 2,RN ), Sε : Lp1(T 2,RN )→ Ck(T 2,RN )

by Sε(f) = φε ∗ f . The proof of the Theorem for maps from Σ = T 2 nowcontinues in exactly the same way as for maps from S1.

Remark 1.5.4. It is interesting to consider Sobolev spaces of maps which donot lie in “Sobolev range.” Thus we could consider

W p1 (Σ,M) = f ∈ Lp1(Σ,M) : f(p) ∈M for almost all p ∈ Σ ,

Hp1,S(Σ,M) = (Closure of C∞(Σ,M) in Lp1(Σ,M)),

for p ≤ dim Σ. Although Hp1,S(Σ,M) ⊆ W p

1 (Σ,M) the inclusion is often strict,and neither space is in general homotopically equivalent to the space C0(Σ,M)of continuous maps. These Sobolev spaces have been extensively studied byHang and Lin [33], among others, and applications to the theory of harmonicmaps are described in the review article of Brezis [11]. When the dimension ofΣ is at least three, harmonic maps from Σ to M are vastly more complicatedthan geodesics and harmonic surfaces; for example, they need not be smooth.

1.6 The α-Lemma*

We have seen that there is a covariant functor from finite-dimensional smoothmanifolds and smooth maps to infinite-dimensional smooth manifolds and smoothmaps,

M 7→ Ck(Σ,M), g : M → N 7→ ωg : Ck(Σ,M)→ Ck(Σ, N),

where ωg(f) = g f . One might hope that when M is a fixed smooth manifold,there is a similar contravariant functor from compact manifolds Σ to infinite-dimensional smooth manifolds Ck(Σ,M) in which

f : Σ1 → Σ2 7→ αf : Ck(Σ2,M)→ Ck(Σ1,M),

where αf (g) = g f . However, it turns out that the maps αf are no longer C∞

smooth, but only Ck:

α-Lemma 1.6.1. If M is a fixed smooth manifold, a smooth map g : Σ1 → Σ2

between smooth compact manifolds induces a Ck map

αg : Ck(Σ2,M)→ Ck(Σ1,M),

34

for each k.

A proof of this lemma can be found in [1].

We could try to put the α- and ω-Lemmas into a single theorem. This is partiallyaccomplished by the following theorem, which we will not prove; it is stated inthe survey article [16], which also includes numerous references:

Theorem 1.6.2. If S, M and N are smooth finite-dimensional manifolds, then

Φ : Ck(S,M)× Ck+s(M,N)→ Ck(S,N), Φ(f, g) = g f,

is Cs.

Remark. The loss of derivatives in the statement of Theorem 1.6.2 has far-reaching implications. For example, suppose that we want to construct examplesof infinite-dimensional Banach Lie groups, which are defined just like ordinaryLie groups, except that of being finite-dimensional manifolds, they are infinite-dimensional Banach manifolds. We could start with a finite-dimensional Liegroup G with identity e, smooth multiplication map

µ : G×G→ G, µ(σ, τ) = σ · τ

and smooth inverse map

ν : G→ G, ν(σ) = σ−1.

We could then define the corresponding loop group L21(S1, G) with identity the

constant map to e, smooth multiplication

ωµ : L21(S1, G)× L2

1(S1, G)→ L21(S1, G), ωµ(f, g) = f · g,

where the dot on the right denotes multiplication within G, and smooth inverse

ων : L21(S1, G)→ L2

1(S1, G), ων(f)(p) = (f(p))−1.

It is not difficult to check that L21(S1, G) is in fact an infinite-dimensional Banach

Lie group. A rich theory of these loop groups has been developed, although theyare often modeled on Frechet rather than Banach spaces.

On the other hand, although the space of Ck diffeomorphisms is a per-fectly well-behaved topological group under composition, this composition failsto be smooth because of the loss of derivatives implicit in the statement of theTheorem 1.6.2. There seems to be no simple way to make the group of Ck

diffeomorphisms into an infinite-dimensional Lie group modeled on a Banachspace. This fact seems to interfere with potential applications of global analysistechniques to important nonlinear systems of PDE’s, such as those governingincompressible fluids.

One consequence of Theorem 1.6.2 is a smoothness result for the evaluation map

ev : Ck(Σ,M)×M →M defined by ev(f, p) = f(p). (1.9)

35

Lemma 1.6.3.The map ev : Ck(Σ,M)× Σ→M , defined by (1.9).

For a direct proof we refer to [2], page 99.

We can extend Theorem 1.6.2 and many of its consequences to the Sobolev man-ifolds Lpj (Σ2,M) when p and k are large enough that Lpj (Σ2,M) ⊂ Ck(Σ2,M)in accordance with (1.7). For example, one consequence is:

Lemma 1.6.4. If Lpj (Σ,M) ⊂ Ck(Σ,M), the map

ev : Lpj (Σ,M)× Σ→M defined by ev(f, p) = f(p)

is Ck.

Note that it follows from this Lemma that

ev : L2k(S1,M)× Σ→M is Ck−1,

while if Σ is a surface,

ev : Lpk(Σ,M)× Σ→M is Ck−1,

when p > 2.

1.7 The tangent and cotangent bundles

Many constructions from the theory of finite-dimensional manifolds can be gen-eralized to infinite-dimensional Banach or Hilbert manifolds. These includetensors of various ranks, vector fields and differential equations, connections,Riemannian metrics on Hilbert manifolds, Finsler metrics on Banach mani-folds, differential forms and de Rham cohomology. Many of these constructionsare carried out in great detail in the Lang’s book [43] on infinite-dimensionalmanifolds. We provide a brief summary here.

We first extend familiar definitions of tangent and cotangent bundles to theinfinite-dimensional context. LetM be an infinite-dimensional smooth manifoldmodeled on a Banach space E with smooth atlas Uα, φα) : α ∈ A. Considerthe collection of triples (α, p, v), where α ∈ A, p ∈ Uα and v ∈ E. On thiscollection of triples we define an equivalence relation ∼ by

(α, p, v) ∼ (β, q, w) ⇔ p = q and w = D(φβ φ−1α )(φα(p))v.

The set of equivalence classes is called the tangent bundle of M and is denotedby TM.

Let [α, p, v] denote the equivalence class of (α, p, v) and define

π : TM−→M by π([α, p, v]) = p.

Let Uα = [α, p, v]; p ∈ Uα, v ∈ E, and define

φα : Uα −→ E ⊕ E by φα([α, p, v]) = (φα(p), v).

36

Then (Uα, φα) : α ∈ A is a smooth atlas on TM making TM into a smoothmanifold modeled on the Banach space E⊕E. If p ∈M, we let TpM = π−1(p),the fiber of the tangent bundle over p, and call TpM the tangent space to Mat p.

Just as in the finite-dimensional case, elements of TpM are called tangentvectors. If γ : (a, b)→M is a smooth curve, t ∈ (a, b) and γ(t) ∈ Uα, we define

γ′(t) ∈ TpM by γ′(t) = [α, γ(t), D(φα γ)(t) · 1],

a tangent vector called the velocity vector to γ at t.If F :M→ N is a smooth map between manifolds with atlases (Uα, φα) :

α ∈ A and (Vβ , ψβ) : β ∈ B and p ∈ M, we can define the differential(F∗)p : TpM→ TF (p)N by

(F∗)p([α, p, v]) = (β, F (p), (D(ψβ F φ−1α )(φα(p))(v)],

where p ∈ Uα and F (p) ∈ Vβ . Note that if γ : (a, b)→M is a C1 curve,

(F∗)p(γ′(t)) = (F γ)′(t), for t ∈ (a, b).

The differentials fit together to form a map of tangent bundles F∗ : TM→ TN .In a very similar way, we can also describe the cotangent bundle of M. We

consider a similar collection of triples (α, p, v∗), where α ∈ A, p ∈ Uα andv∗ ∈ E∗, where E∗ is the Banach space dual to E. This time we choose theequivalence relation

(α, p, v∗) ∼ (β, q, w∗) ⇔ p = q and v∗ = [D(φβ φ−1α )(φα(p))]∗w∗,

where (·)∗ denotes transpose map defined by([D(φβ φ−1

α )(φα(p))]∗w∗)

(v) = w∗(D(φβ φ−1α )(φα(p))(v).

The cotangent bundle T ∗M is the set of equivalence classes and is a smoothmanifold modeled on the Banach space E⊕E∗. Once again, we have a projection

π : TM−→M defined by π([α, p, v]) = p.

If p ∈M, the fiber T ∗pM of the cotangent bundle over p is called the cotangentspace to M at p. A smooth map F : M → N induces a map in the oppositedirection

(F ∗)p : T ∗F (p)N → TpM by (F ∗)p(w∗)(v) = w∗((F∗)p(v)).

In terms of local coordinates, this can be written

(F ∗)p([β, F (p), w∗]) = [α, p, (D(ψβ F φ−1α )(φα(p))∗(w∗)]. (1.10)

We can also define the k-th tensor power and the k-th exterior power of thecotangent bundle. To do this, we start with the Banach space Lk(E,R) of maps

T : E × E × · · ·E(k times) −→ R

37

which are linear in each variable, the so-called space of k-linear maps, or itssubspace of alternating k-linear maps Lka(E,R). An element T ∈ Lk(E,R) issaid to be alternating if

T (hσ(1), . . . , hσ(k)) = (sgn(σ))T (h1, . . . , hk), for all σ ∈ Sk,

where Sk denotes the symmetric group on k letters and sgn(σ) denotes the signof the permutation σ ∈ Sk. A continuous linear map Φ : E → F betweenBanach spaces induces continuous linear maps

Φ∗ : Lk(F,R)→ Lk(E,R), Φ∗ : Lka(F,R)→ Lka(E,R)

by means of the formula

(Φ∗T )(v1, . . . vk) = T (Φ(v1), . . . ,Φ(vk)).

To define the k-th tensor power of the cotangent bundle, we start with triples(α, p, T ), where α ∈ A, p ∈ Uα and T ∈ Lk(E,R) and the equivalence relation

(α, p, Tα) ∼ (β, q, Tβ) ⇔ p = q and Tα = Φ∗Tβ ,

where Φ = D(φβ φ−1α )(φα(p)). The k-th tensor power of the cotangent bundle

⊗kT ∗M is the set of equivalence classes. The k-th exterior power is definedthe same way, except that T is taken to lie in Lka(E,R). The fibers ⊗kT ∗pMand ΛkT ∗pM are called the k-th tensor power and the exterior power of thecotangent space to M at p.

In the case where M = L21(S1,M), M being oriented, the tangent bundle

has another description, namely

TL21(S1,M) = L2

1(S1, TM).

To see this, recall how we constructed the atlas on L21(S1,M). If γ is a

smooth element of L21(S1,M), we let

Uγ,ε = λ ∈ L21(S1,M) : dM (λ(t), γ(t)) < ε for all t ∈ S1,

Vγ,ε = X ∈ L21(S1,Rn)) : 〈X(t), X(t)〉 < ε for all t ∈ S1,

and define

ψγ,ε : Vγ,ε −→ Uγ,ε by (ψγ,ε(X))(t) = expγ(t)(X(t)).

For ε sufficiently small, ψγ,ε is a bijection with inverse φγ,ε : Uγ,ε → Vγ,ε, and

(Uγ,ε, φγ,ε) : γ is smooth and ε is sufficiently small

is a smooth atlas for L21(S1,M).

Now for each smooth γ, we can construct a lift γ ∈ L21(S1, TM) by setting

γ(t) = 0γ(t) ∈ Tγ(t)M,

38

and construct a corresponding chart on L21(S1, TM). The remarkable fact is

that we can choose the chart to be valid over all of

Uγ,ε = Ω−1π (Uγ,ε) = X ∈ L2

1(S1, TM) : π X ∈ Uγ,ε.

Indeed, we can set

Vγ,ε = Vγ,ε × L21-sections of γ∗TM

and define ψγ,ε : Vγ,ε → Uγ,ε by

ψγ,ε(X,Y ) = (expγ(t)X(t), (d(expγ(t))X(t)Y (t)).

Finally, define φγ,ε : Uγ,ε → Vγ,ε by φγ,ε = ψ−1γ,ε. Then

(φγ1,ε φ−1γ2,ε)(X,Y ) = ((φγ1,ε φ−1

γ2,ε)(X), D(φγ1,ε φ−1γ2,ε)(X)(Y )).

Thus the charts transform exactly the way they should for the tangent bundle.In a quite similar fashion, we can show that if Σ is a compact Riemann

surface and p > 2,TLp1(Σ,M) = Lp1(Σ, TM).

It is important to observe that just as the imbedding i : M → RN induces animbedding ωi : L2

1(S1,M) → L21(S1,RN ), so the imbedding i : TM → TRN =

R2N induces an imbedding

ωi : TL21(S1, TM) = L2

1(S1, TM) −→ L21(S1, TRN ) = L2

1(S1,R2N ),

allowing us to realize TL21(S1, TM) as a subspace of a Banach space. Similarly,

TLp1(Σ,M) can be regarded as a subspace of a Banach space.Note that the Hilbert space inner product allows us to identify the model

space L21(S1,Rn) for M = L2

1(S1,M) with its dual. Using this fact, it is notdifficult to verify that

T ∗L21(S1,M) = L2

1(S1, T ∗M), ⊗kT ∗L21(S1,M) = L2

1(S1,⊗kT ∗M),

andΛkT ∗L2

1(S1,M) = L21(S1,ΛkT ∗M).

It is actually the “sections” of the tangent bundle and the exterior powersof the cotangent bundle that will be of most importance for us; these are calledvector fields and differential forms, respectively.

Definition. A smooth vector field on M is a smooth map

X :M−→ TM such that π X = idM.

Note that if X :M→ TM is a smooth vector field and f :M→ R is a smoothfunction, we can define the vector field fX :M→ TM by (fX )(p) = f(p)X (p).

39

This makes the space of smooth vector fields into a module over the ring ofsmooth real-valued functions.

Example. If M is a finite-dimensional Riemannian manifold and X : M → TMis a smooth vector field on M , then

ωX : L21(S1,M) −→ L2

1(S1, TM) = TL21(S1,M) and

ωX : Lp1(Σ,M) −→ Lp1(Σ, TM) = TL21(Σ,M)

are smooth vector fields on L21(S1,M) and Lp1(Σ,M).

1.8 Differential forms

For calculations on smooth manifolds, differential forms are often more conve-nient to use than general tensor fields. We now describe how some of the familiaroperations on differential forms extend to infinte-dimensional manifolds.

Definition. A smooth covariant tensor field of rank k on M is a smooth map

φ :M−→ ⊗kT ∗M such that π φ = idM.

A smooth differential form of degree k onM (or a smooth k-form) is a smoothmap

φ :M−→ ΛkT ∗M such that π φ = idM.

As in the case of vector fields, we can multiply covariant tensor fields or differ-ential forms by functions.

An important example of differential one-form occurs when f :M→ R is asmooth function. Then for the coordinate chart (Uα, φα), we have

D(f φ−1α ) : Uα → L(E,R) = E∗, (1.11)

where E is the model space ofM. The differential of f is the smooth one-formdf such that

df(p) = [α, p,D(f φ−1α )(p)], for p ∈ Uα.

It is readily verified that the local representatives transform as they shouldunder change of coordinates. Moreover it is easily checked that the differentialsof functions at a point p generate the cotangent space. The Leibniz rule fordifferentiation implies that d(fg) = gdf + fdg.

Definition. A point p ∈ M is a critical point for the real-valued functionf :M→ R if df(p) = 0.

An important example to keep in mind is the real-valued function

J : L21(S1,M)→M, J(γ) = J(γ) =

12

∫S1|γ′(t)|2dt,

40

when M is a Riemannian manifold. In this case, regularity theory will showthat a critical point in this case is actually a C∞ map, hence a smooth closedgeodesic in M .

Using the notion of differential of a function, we can define the directionalderivative of a function f in the direction of X by

X (f)(p) = df(p)(X (p)),

the right hand side being the dual pairing between cotangent and tangent spaces.

Lemma 1.8.1. Let X and Y be smooth vector fields on the Banach manifoldM. Then there is a unique vector field [X ,Y] onM which satisfies the equation

([X ,Y](f))(p) = (XY(f))(p)− (YX (f))(p).

Sketch of proof: It suffices to show that the above formula is equivalent to anexpression for [X ,Y] in terms of a local coordinate chart (Uα, φα). Suppose that

X , Y : Uα → E are defined by X (p) = [p, α, X (p)], Y(p) = [p, α, Y(p)].

Using the chain rule, one can check that the Lie bracket must then be given by

[X ,Y](p) =[p, α,DY(p)X (p)−DX (p)Y(p)

].

The vector field [X ,Y] is known as the Lie bracket of X and Y; it is easilyverified that it satisfies the identity:

[fX , gY] = fg[X ,Y] + fX (g)Y − gY(f)X .

We next note that differential forms are “functorial.” If F : M → N is asmooth map and φ is a smooth differential form of degree k on N , we can definea differential form F ∗φ on M by

[F ∗φ](p) = F ∗p (φ(F(p));

it follows from (1.10) that F ∗φ is smooth.

If X1, . . . , Xk are smooth vector fields on M and φ is a smooth k-form on M,then the smooth function

φ(X1, . . . ,Xk) :M−→ R

is defined fiberwise via the continuous (k + 1)-linear map

Lka(E,R)× E × · · · × E −→ R,

where E is the model space for M.

41

Definition. If φ is a smooth k-form onM and ω is a smooth l-form onM, thewedge product of φ and ω is the (k + l)-form on M defined by

(φ ∧ ω)(X1, . . . ,Xk+l) =k!l!

(k + l)!

∑σ∈Sk+l

sgn(σ)(Xσ(1), . . . ,Xσ(k+l)).

Here Sk+l is the symmetric group on k + l letters and sgn(σ) is the sign of thepermutation σ ∈ Sk+l.

We remind the reader that some authors prefer to define the wedge productusing the factor

1(k + l)!

instead ofk!l!

(k + l)!.

With either convention, the wedge product is bilinear and associative, just as inthe case of finite-dimensional manifolds, but not commutative. If φ is a k-formand ω an l-form,

φ ∧ ω = (−1)klω ∧ φ.

Definition. If φ is a smooth k-form on M and X is a smooth vector field, theinterior product ιXφ is the smooth (k − 1)-form on M defined by the formula

ιXφ(X2, . . . ,Xk) = φ(X ,X2, . . . ,Xk),

whenever X1, . . . , Xk are smooth vector fields onM. (It is readily checked thatthere is a unique such differential form.) It is easily checked that

ιX (φ ∧ ψ) = (ιXφ) ∧ ψ + (−1)deg(φ)φ ∧ ιXψ.

Finally, the exterior derivative d is the collection of R-linear maps fromk-forms to (k+1)-forms which satisfy the following axioms, familiar from finite-dimensional differential topology:

1. If ω is a k-form, the value dω(p) depends only on ω and its derivatives atp.

2. If f is a smooth real-valued function regarded as a differential 0-form, d(f)is the differential of f defined before.

3. d d = 0.

4. If ω is a k-form and φ is an l-form, then

d(ω ∧ φ) = (dω) ∧ φ+ (−1)kω ∧ (dφ).

5. If F : N →M is a smooth map, F ∗ d = d F ∗ on differential forms.

42

Just as in the finite-dimensional case, one can prove:

Theorem 1.8.2. There is a unique of linear maps of real vector spaces,

d : differential k-forms −→ differential (k + 1)-forms,

which satisfy the five above axioms. Moreover, these linear maps satisfy theexplicit formula

dω(X0, . . . ,Xk) =∑

(−1)iXi(ω(X0, . . . , Xi, . . . ,Xk)

)+∑i<j

(−1)i+jω([Xi,Xj ],X0, . . . , Xi, . . . , Xj , . . . ,Xk), (1.12)

where the hats denote elements which are left out.

We sketch the proof under the assumption that the corresponding theorem forfinite-dimensional manifolds has been established. Using the fifth axiom, wecan reduce the proof of uniqueness to the case where the manifold is an opensubset of the model space. Since (1.12) is linear over functions, it suffices toestablish the formula in the case where X0, . . .Xk are constant in terms ofthe local chart, in which case all brackets [Xi,Xj ] vanish. Thus it suffices toprove uniqueness when the vector fields X0, . . .Xk are tangent to a (k + 1)-dimensional affine subspace of the model space, and in this case (1.12) followsfrom the corresponding formula on this (k + 1)-dimensional subspace, a finite-dimensional submanifold.

To prove the local existence, one merely defines the exterior derivatives bythe formula (1.12) and check that they satisfy all of the axioms. It is simplestto verify the first four axioms by showing that if any of these axioms were tofail, it would have to fail already on finite-dimensional manifolds.

Finally, one notes that if operators satisfying the five axioms are defined andunique for any open subset of the model space, they are defined and unique forany set Uα in a smooth atlas Uα : α ∈ A for M. These operators must agreeon overlaps Uα ∩ Uβ , for α, β ∈ A, and hence must fit together to form well-defined operators on M, and the resulting operators must be unique becausetheir restrictions to each Uα are unique.

Differential forms of degree k should be thought of as integrands for inte-gration over compact oriented k-dimensional submanifolds. From this pointof view, it is evident that they should be determined by their restrictions tofinite-dimensional manifolds, where the above axioms for exterior derivative arefamiliar.

1.9 Riemannian and Finsler metrics

To construct critical points of functions such as the action or energy, we needto develop the “method of steepest descent” within the context of infinite-dimensional manifolds. And to find the direction of steepest descent, we are

43

led to seek a notion of gradient, which in the finite-dimensional context de-pends upon a Riemannian metric. Thus it is important to consider how toextend the notion of Riemannian metric to infinite-dimensional manifolds. It isto be expected that a somewhat stronger theory is possible for Hilbert manifoldsthan for Banach manifolds. Indeed, we will see that there is no fully satisfactorynotion of Riemannian metric or gradient on Banach manifolds. We will need tomake do with weaker notions of Finsler metrics and pseudogradients.

Suppose, therefore that M is a Hilbert manifold modeled on the Hilbertspace E. If (Uα, φα) is a smooth chart on M and Uα is the subset of TMprojecting to Uα, define

εα : Uα −→ E by εα([α, p, v]) = v.

The Hilbert space inner product (·, ·) pulls back via εα to TpM: we let

(·, ·)α : TpM× TpM→ R by (v, v)α = (εα(v), εα(w)).

Definition. A Riemannian metric on a Hilbert manifoldM is a function whichassigns to each p ∈M an inner product

〈·, ·〉p : TpM× TpM−→ R

such that:

1. There is some constant cp > 0 such that

1cp

(v, v)α < 〈v, v〉p < cp(v, v)α, for all v ∈ TpM.

(Thus the topology induced by the Riemannian metric on TpM agreeswith the model space topology.)

2. 〈·, ·〉p varies smoothly with p; in other words, p 7→ 〈·, ·〉p is a smoothcovariant tensor field of rank two.

Example 1.9.1. Suppose that we have a proper isometric imbedding of thesmooth Riemannian manifold M into RN . Then the Hilbert manifold L2

1(S1,M)can be given a very natural Riemannian metric 〈〈·, ·〉〉 as follows: if X,Y ∈TγL

21(S1,M), we can regard X and Y as maps X,Y : S1 → RN such that

X(t) ∈ Tγ(t)M for each t ∈ S1. We set

〈X,Y 〉γ =∫S1

[X(t) · Y (t) +X ′(t) · Y ′(t)]dt.

This can be regarded as the pullback of the “flat” Riemannian metric on theHilbert space L2

1(S1,RN ), and it is smooth because the pullback of a smoothcovariant tensor field via a smooth map is smooth.

44

Definition. IfM is a Hilbert manifold with Riemannian metric p 7→ 〈·, ·〉p, thegradient of a C1 function f :M→ R is the vector field grad(f) defined by

〈grad(f)(p), v〉p = dfp(v), for all v ∈ TpM.

The idea behind the method of steepest descent for finding critical points of anonnegative function f is to follow flowlines for the vector field −grad(f); infavorable cases, these flowlines will converge to a critical point for f .

In the case of Banach manifolds, the metrics best suited to our applicationsare not Riemannian, but Finsler. If (Uα, φα) is a smooth chart on a Banachmanifold M, Uα is the subset of TM projecting to Uα and

εα : Uα −→ E by εα([α, p, v]) = v

as before, the Banach space norm ‖ · ‖ on E pulls back via εα to TpM: we let

‖ · ‖α : TpM→ R by ‖v‖α = ‖εα(v)‖.

Definition. A Finsler metric on a Banach manifold M is a function whichassigns to each p ∈M a norm

‖ · ‖p : TpM−→ R

such that

1. There is some constant cp > 0 such that

1cp‖v‖α < ‖v‖p < cp‖v‖α, for all v ∈ TpM.

(Thus the Finsler norm on TpM is equivalent to the Banach space normfor the model space.)

2. ‖ · ‖p varies continuously with p.

Note that any Riemannian metric on a Hilbert manifold determines a Finslermetric: simply define

‖ · ‖p : TpM−→ R by ‖v‖p =√〈v, v〉p.

Even in this special case, however, the norm ‖·‖p is only continuous, not smoothas a function on TM.

The Riemannian metric on a Hilbert manifold establishes a norm-preservingvector bundle isomorphism from TpM to T ∗pM. We do not have such an iso-morphism in the case of a Finsler metric on a Banach manifold, but the norm‖·‖p on TpM induces a dual norm (which we also denote by ‖·‖p for simplicity)on T ∗pM:

‖φ‖p = sup|φ(v)| : v ∈ TpM and ‖v‖p = 1.

45

Example 1.9.2. Suppose that we have a proper isometric imbedding of thesmooth Riemannian manifold M into RN . Given a compact Riemann surface Σand a real number p > 2, the Banach manifold Lp1(Σ,M) can be given a Finslermetric as follows: if X ∈ Tf (Lp1(Σ,M), we can regard X as a map X : Σ→ RNsuch that X(p) ∈ Tf(p)M for each p ∈ Σ. We then let ‖X‖f be the Lp1-normof X as a mapping into Euclidean space. The Finsler metric f 7→ ‖ · ‖f canbe regarded as the pullback of the “flat” Finsler metric on the Banach spaceLp1(Σ,RN ).

IfM is a connected Banach manifold with Finsler metric ‖·‖ and γ : [0, 1]→Mis a C1 curve, we can define its length L(γ) by

L(γ) =∫ 1

0

‖γ′(t)‖dt,

where integration along a path can be defined as in [43], §1.4. We can thendefine a distance function d :M×M→ R by

d(p, q) = infL(γ) : γ : [0, 1]→M is a C1 path, γ(0) = p, γ(1) = q

. (1.13)

Proposition 1.9.3. Given a Finsler metric on a regular Banach manifold, thedistance function d defined aboveis a metric in the metric space sense, and themetric topology agrees with the manifold topology.

We apologize to the reader for not giving a complete proof of this Proposition,referring instead to [61] (see the appendix to §2).

However we will describe a proof of the Proposition for Examples 1.8.1 and1.8.2. Note that it is quite easily verified that the distance function d satisfies

d(p, q) = d(q, p), d(p, r) ≤ d(p, q) + d(q, r), and d(p, p) = 0.

That only leaves the property d(p, q) = 0⇒ p = q.

Lemma 1.9.4. In each of our two key examples, L21(S1,M) and Lp1(Σ,M) with

p > 2, d is a metric and the metric topology agrees with the manifold topology.Moreover, L2

1(S1,M) and Lp1(Σ,M) are complete as metric spaces.

Proof: Let us consider Lp1(Σ,M). If f, g ∈ Lp1(Σ,M) and d(f, g) = 0, thenthere exist arbitrarily short paths connecting f and g in Lp1(Σ,M). But a pathconnecting f and g in Lp1(Σ,M) also connects f and g in the ambient Banachspace E = Lp1(Σ,RN ), so there are arbitrarily short curves connecting f andg in the ambient Banach space. However, by a straightforward modification ofthe finite-dimensional argument, it is easily verified that if γ : [0, 1] → E is aC1-map into a Banach space, then∫ 1

0

‖γ′(t)‖dt ≥ ‖γ(0)− γ(1)‖,

46

so two distinct points in E cannot be joined by curves of arbitrarily small length.Since the map ωi : Lp1(Σ,M)→ Lp1(Σ,RN ) induced by the inclusion i : M →

RN is an imbedding, it is now easy to verify that the metric topology agrees withthe manifold topology. Finally, since ωi : Lp1(Σ,M) → Lp1(Σ,RN ) is distance-decreasing, a Cauchy sequence fi in Lp1(Σ,M) is also a Cauchy sequence inLp1(Σ,RN ), and must therefore converge. Since Lp1(Σ,M) is a closed subset ofLp1(Σ,RN ), we see that Lp1(Σ,M) must be complete as a metric space.

The case of L21(S1,M) is treated in the same way.

1.10 Vector fields and ODE’s

It is well-known that the global qualitative theory of systems of ordinary dif-ferential equations is best formulated within the language of vector fields onfinite-dimensional manifolds. This theory, including the fundamental existenceand uniqueness theorem for systems of ordinary differential equations, can beextended to infinite-dimensional manifolds. A detailed exposition of this exten-sion is presented in Chapter IV of [43].

Definition. A C1 curve γ : (a, b)→M is called an integral curve for the vectorfield X if

X (γ(t)) = γ′(t), for t ∈ (a, b), (1.14)

where γ′(t) is the velocity vector to γ at t.

Just as in the finite-dimensional case, a fundamental existence and uniquenesstheorem states that given a smooth vector field X on M, there is a uniqueintegral curve for X which passes through any point of M:

Theorem 1.10.1. (Existence and Uniqueness Theorem for OrdinaryDifferential Equations.) Suppose that X is a C1 vector field on M andp ∈M. Then there is an open neighborhood U of p, an ε > 0 and a C1 map

φ : (−ε, ε)× U −→M

such that if φt(q) = φ(t, q) for t ∈ (−ε, ε) and q ∈ U , then

1. each curve t 7→ φt(q) is an integral curve for X ,

2. any integral curve for X which passes through U is of the form t 7→ φt(q)for some q ∈ U ,

3. φ0 is the inclusion U ⊂M, and

4. φt φs = φt+s, whenever, both sides are defined.

We will call (−ε, ε)× U a local flow box for X .

Idea of proof (following Chapter IV of Lang [43]): The proof is based upon theContraction Lemma, just like the proof of the Inverse Function Theorem.

47

We can replace the differential equation (1.14) by its local coordinate repre-sentation, and consider the initial value problem

γ′(t) = f(γ(t)), γ(0) = q, (1.15)

for γ : (a, b)→ V and f : V → E, where V is a suitable open subset of a Banachspace E. We can assume that

‖f‖ ≤ K and ‖Df‖ ≤ L

on V . Integrating both sides of (1.15) yields the equivalent integral equation

γ(t) = q +∫ t

0

f(γ(u))du. (1.16)

We can assume that the closed ball B2δ(p) of radius δ about p is contained in Vand suppose that q ∈ Bδ(p), the open ball of radius δ about p. Let I = [−ε, ε],where ε > 0 will be chosen later, and let

X =γ : I → U : γ is continuous, γ(0) = q and γ(I) ⊂ B2δ(p)

.

We can make X into a complete metric space by defining the distance functiond by

d(γ1, γ2) = sup‖γ1(t)− γ2(t)‖ : t ∈ I.If γ ∈ X, we set

T (γ)(t) = q +∫ t

0

f(γ(u))du.

We choose ε so that ε < δ/K, and hence

‖T (γ)(t)− q‖ ≤ εK ≤ δ,

so T (γ) ∈ X. Finally, we note that

d (T (γ1), T (γ2)) = sup ‖T (γ1)(t)− T (γ2)(t)‖ : t ∈ I≤ ε sup‖f(γ1(t))− f(γ2(t))‖ : t ∈ I

≤ εL sup‖γ1(t)γ2(t)‖ : t ∈ I = εLd(γ1, γ2).

Thus by choosing ε so that εL < 1, we can ensure that T : X → X will bea contraction. Then, by the Contraction Lemma, T has a unique fixed pointγq ∈ X, which must be a solution to the integral equation (1.16). This fixedpoint γq is C1 and is the unique solution to the initial value problem (1.15).

Thus we can define

φ : (−ε, ε)×Bδ(p)→ V by φ(t, q) = γq(t).

It is relatively easy to check that properties 2, 3 and 4 of the Theorem hold andthat φ is continuous. It is a little more challenging to check that φ is C1, andfor that we refer the reader to the excellent presentation in [43].

48

Once we have the Existence and Uniqueness Theorem, we can piece togetherthe locally defined maps to form a map

φ : V −→M, where V is an open neighborhood of 0 ×M in R×M.

We say that the maps φt defined by φt(q) = φ(t, q) form the one-parametergroup of local diffeomorphisms of M which corresponds to the vector field X .

1.11 Condition C

We want to apply the existence and uniqueness theorem from the precedingsection to find critical points of a C2 real-valued map f : M→ [0,∞) via themethod of steepest descent, where M is an infinite-dimensional manifold. Inorder to get this method to work, we need to assume that the function f assumesa “compactness condition” introduced by Palais and Smale [62]:

Definition. Suppose that M is a Banach manifold with a complete Finslermetric. (For example,M might be a Hilbert manifold with a complete Rieman-nian metric.) Then a C2 function f :M→ [0,∞) is said to satisfy condition Cif whenever pi is a sequence in M such that

1. f(pi) is bounded and

2. ‖df(pi)‖ is not bounded away from zero,

then pi possesses a subsequence which converges to a critical point for f .

It is easiest to utilize this compactness condition in the case where M is aHilbert manifold:

Theorem 1.11.1. Suppose that M is a Hilbert manifold with a completeRiemannian metric 〈·, ·〉. If f : M → [0,∞) is a C2 function which satisfiescondition C, X = −grad(f) and φt is the local one-parameter group of dif-feomorphisms corresponding to X , then

1. for each p ∈M, φt(p) is defined for all t ≥ 0, and

2. there is a sequence ti →∞ such that φti(p) converges to a critical pointfor f .

The proof will be based upon a collection of inequalities. Suppose that 0 < t1 <t2. Then

f(φt1(p))− f(φt2(p)) = −∫ t2

t1

d

dtf(φt(p))dt

= −∫ t2

t1

df(φt(p))(X (φt(p))dt =∫ t2

t1

〈grad(f)(φt(p)),X (φt(p)〉dt

=∫ t2

t1

‖grad(f)(φt(p))‖2dt =∫ t2

t1

‖df(φt(p))‖2dt. (1.17)

49

On the other hand, using the metric d on the Riemannian manifoldM, we have

d(φt1(p), φt2(p)) ≤∫ t2

t1

∥∥∥∥ ddt (φt(p))∥∥∥∥ dt

≤∫ t2

t1

‖X (φt(p))‖dt =∫ t2

t1

‖df(φt(p))‖dt. (1.18)

Now we use the Cauchy-Schwarz inequality to obtain

d(φt1(p), φt2(p))2 ≤[∫ t2

t1

‖df(φt(p))‖dt]2

≤ (t2 − t1)∫ t2

t1

‖df(φt(p))‖2dt = (t2 − t1)(f(φt1(p))− f(φt2(p))). (1.19)

Let t = supt ∈ R : φt(p) is defined, and let ti be a sequence of realnumbers < t such that ti → t. It follows from (1.19) that φti(p) is a Cauchysequence in M. Since (M, d) is a complete metric space, φti(p) → q, for someq ∈ M. But by the Existence and Uniqueness Theorem there is a flow box(−ε, ε) × U containing (0, q). This implies that the curve t 7→ φt(p) can beextended beyond t, giving a contradiction. Thus we see that φt(p) is defined forall t ≥ 0, and the first statement of the Theorem is proven.

Next, it follows from (1.17) that∫ ∞0

‖grad(f)(φt(p))‖2dt <∞,

and hence there must exist a sequence ti →∞ such that

‖df(φti(p))‖ = ‖grad(f)(φti(p))‖ → 0.

By Condition C, a subsequence of φti(p) converges to a critical point for f ,finishing the proof of the Theorem.

Of course, we would like a version of the above Theorem to hold for the caseof a C2 function f : M → R, where M is only a Banach manifold. However,it is not possible to define Riemannian metrics on M in this case, so we needa replacement for the notion of gradient, such as the following, similar to adefinition suggested by Palais [59]:

Definition. Suppose that f : M→ R is a C2 function on a Banach manifoldwhich has a Finsler metric and let U be an open subset ofM. A C1 vector fieldX : U → TM is called a pseudogradient for f over U if there is a constant ε > 0such that for each p ∈ U ,

1. dfp(X (p)) ≥ ε‖dfp‖2,

2. ‖X (p)‖ ≤ (1/ε)‖dfp‖.

50

In both inequalities, ‖dfp‖ = sup|dfp(v)| : v ∈ TpM and ‖v‖ ≤ 1, which is thedual norm on the cotangent space to M at p.

Of course, in the case of a Hilbert manifold, X = grad(f) satisfies both con-ditions in the definition with ε = 1; in other words, a gradient on a Hilbertmanifold is also a pseudogradient. The definition was set up so that the follow-ing Theorem would be true:

Theorem 1.11.2. Suppose that M is a Banach manifold with a completeFinsler metric ‖ · ‖. Suppose that f : M → [0,∞) is a C2 function whichsatisfies condition C. Let

K = p ∈M : df(p) = 0

and let U =M−K. If X is a pseudogradient for f on U , and φt is the localone-parameter group of diffeomorphisms corresponding to −X , then

1. for each p ∈M, φt(p) is defined for all t ≥ 0, and

2. there is a sequence ti →∞ such that φti(p) converges to a critical pointfor f .

The proof is almost identical to the proof of Theorem 1.11.1 (except that wecall the vector field −X instead of X ). Assuming that p is not a critical pointfor f and that 0 < t1 < t2, we use the first condition in the definition ofpseudogradient to replace (1.17) by the inequality

f(φt1(p))− f(φt2(p)) = −∫ t2

t1

d

dtf(φt(p))dt

=∫ t2

t1

df(φt(p))(X (φt(p))dt ≥ ε∫ t2

t1

‖df(φt(p))‖2dt. (1.20)

We use the second condition in the definition of pseudogradient to replace (1.18)by

d(φt1(p), φt2(p)) ≤∫ t2

t1

‖X (φt(p))‖dt ≤1ε

∫ t2

t1

‖df(φt(p))‖dt. (1.21)

Then we use the Cauchy-Schwarz inequality exactly as before to obtain

d(φt1(p), φt2(p))2 ≤ 1ε2

[∫ t2

t1

‖df(φt(p))‖dt]2

≤ t2 − t1ε2

∫ t2

t1

‖df(φt(p))‖2dt ≤t2 − t1ε3

(f(φt1(p))− f(φt2(p))). (1.22)

We can now use (1.22) instead of (1.19) to show that φt(p) is defined for allt ≥ 0. Finally, it follows from (1.20) that∫ ∞

0

‖df(φt(p))‖2dt <∞,

51

which enables us to find a sequence ti →∞ such that ‖df(φti(p))‖ → 0, and bycondition C, a subsequence of φti(p) converges to a critical point for f .

The only problem now is to show that given a C2 function f : M → [0,∞)on a Banach manifold, we can construct a corresponding pseudogradient onU =M−K, where K is the set of critical points for f . The standard techniquefor constructing a pseudogradient consists of constructing pseudogradients overeach open set of an open cover of U and then piecing these together using apartition of unity.

Let M be a metrizable infinite-dimensional smooth manifold modeled ona Banach space. According to a well-known theorem of Stone, M must beparacompact. That means that every open cover of M must have an openlocally finite refinement.

Definition. Let U = Uα : α ∈ A be an open cover of M. A partition ofunity subordinate to U is a collection ψα : α ∈ A of continuous real-valuedfunctions on M such that

1. ψα :M→ [0, 1],

2. the support of ψα is a closed subset of Uα,

3. if p ∈ M, there is an open neighborhood V of p which intersects thesupports of only finitely many ψα, and

4.∑ψα = 1.

It is known (and proven in topology texts) that any open cover of a paracompactHausdorff space possesses a subordinate continuous partition of unity.

Moreover, as proven in Lang [43], C∞ Hilbert manifolds possess C∞ parti-tions of unity. However, for Banach manifolds, we encounter a perhaps unex-pected obstacle. Banach manifolds need not possess C∞ partitions of unity. Aspointed out in [16], for example, to construct Ck partitions of unity one needsto be able to construct nontrivial real-valued Ck functions on the model Banachspace E with bounded support.

Fortunately, in the case where Σ is a Riemann surface and p > 2 the Banachmanifold Lp1(Σ,M) does possess partitions of unity of class C2. To see why, wenotice that the function

f : Lp1(Σ,R) −→ R defined by f(φ) = ‖φ‖p

is C2, by an argument similar to that given in Example 1.2.3. Let g : R→ [0, 1]be a smooth function such that

1. g(s) = 1 when |s| ≤ 1, and

2. g(s) = 0 when |s| ≥ 2.

52

Then the map

fε : Lp1(Σ,Rn)→ [0, 1] defined by fε(φ) = g

(2f(φ)ε

)is a C2 function which equals one on a small neighborhood of the origin andhas support contained in the set φ ∈ Lp1(Σ,Rn) : ‖φ‖ ≤ ε. Using localcoordinates, we can transport this function to Lp1(Σ,M) thereby obtaining a C2

function f : Lp1(Σ,M) → R which is one in a neighborhood of a given point pand vanishes outside a given open neighborhood of p.

Using this function to start with, we can follow the familiar argument (suchas given in Lang [43], Chapter II, §3) to construct C2 partitions of unity subor-dinate to any open cover on the smooth manifold Lp1(Σ,M).

Lemma 1.11.3. If f : Lp1(Σ,M)→ R is a C2 function, where p ≥ 2, then f pos-sesses a C2 pseudogradient X which is tangent to every Lpk(Σ,M) ⊂ Lp1(Σ,M),for k ∈ N.

Proof: Suppose that 0 < ε < 1. If p is not a critical point for f , we can choosea unit vector u ∈ TpM such that |dfp(u)| >

√ε‖dfp‖; then

v =√ε‖dfp‖u satisfies ‖v‖ ≤ ‖dfp‖, dfp(v) ≥ ε‖dfp‖2,

the two conditions in the definition of pseudogradient at the point p. We canextend v to a smooth vector field on some neighborhood of p which is a pseudo-gradient; for example, we could choose it to be constant in terms of some smoothchart. Thus we can construct a pseudogradient on an open neighborhood aboutany point p which is not in the set K of critical points of F . If M admits C2

partitions of unity, one can piece together a C2 pseudogradient on M−K.In the above construction, we can choose v to lie in the dense subspace

Lpk(Σ,M) of Lp1(Σ,M) and the C2 partition of unity on Lp1(Σ,M) can be chosenso that it restricts to a C2 partition of unity on Lpk(Σ,M) for every k ≥ 1. Whenthis is done, the pseudogradient will be tangent to every Lpk(Σ,M), finishing theproof of the Lemma.

1.12 Topological constraints give critical points

Suppose now that M is a Banach manifold with a complete Finsler metric.Suppose that f :M→ [0,∞) is a C2 function and let

Ma = p ∈M : f(p) ≤ a.

Definition. An ambient isotopy ofM is a smooth map Ψ :M×[0, 1]→M suchthat if ψt :M→M is defined by ψt(p) = Ψ(p, t), then ψt is a diffeomorphismfor each t ∈ [0, 1] and ψ0 = id.

Theorem 1.12.1. (Deformation Theorem) Suppose that M is a Banachmanifold with a complete Finsler metric. If f : M → R is a nonnegative C2

53

function satisfying condition C and there are no critical points p for f such thata ≤ f(p) ≤ b, then

1. Ma is a strong deformation retract of Mb.

2. there is a smooth ambient isotopy Ψ = ψt : t ∈ [0, 1] of M such thatψ1(Mb) ⊂Ma.

Proof: It follows from Condition C that there is an ε > 0 such that there are nocritical points p for f such that a− ε < f(p) < b+ ε. Using a partition of unity,we construct a smooth function η :M→ [0, 1] such that

1. η ≡ 1 on p ∈M : a ≤ f(p) ≤ b,

2. η ≡ 0 outside p ∈M : a− ε < f(p) < b+ ε.

Let Y be a pseudogradient for f on U =M−K, where K is the critical locusof F and set X = −ηY.

Since f satisfies Condition C, there is a constant k > 0 such that ‖df‖ ≥ kon p ∈M : a ≤ f(p) ≤ b. Indeed, if not, there would exist a sequence pi inp ∈M : a ≤ f(p) ≤ b such that ‖df(pi)‖ → 0. By condition C, a subsequenceof pi would converge to a critical point for f in p ∈ M : a ≤ f(p) ≤ b,contradicting the hypothesis of the Theorem.

Theorem 1.10.2 shows that the one-parameter group φt : t ∈ R of localdiffeomorphisms determined by X is globally defined for all t ≥ 0. We claimthat if p ∈ Mb, then φt(p) ∈ Ma for t > (b − a)/(εk). Indeed, if φt(p) /∈ Ma,it follows from (1.20) that

f(p)− f(φt(p)) ≥ ε∫ t

0

‖df(φτ (p))‖dτ ≥ εkt,

and hence εkt ≤ b − a, or equivalently, t ≤ (b − a)/(εk). We now set ψt = φct,where c = 2(b− a)/(εk), and define

Ψ :M× [0, 1]→M by Ψ(p, t) = ψt(p).

Then Ψ is an ambient isotopy such that ψ1(Mb) ⊂Ma.This proves the second assertion of the Theorem. For the first, we let τ(p)

be the first time t such that φt(p) ∈Ma and define Φ :Mb × [0, 1]→Mb by

Φ(p, t) =

φt(τ(p)), for p ∈Mb −Ma,

p, for p ∈Ma.

Then Φ is a strong deformation retraction from Mb to Ma, finishing the proofof the Theorem.

Definition. Let F be a family of subsets of M. We say that F is ambientisotopy invariant if

A ∈ F ⇒ ψ1(A) ∈ F

54

whenever ψt : t ∈ [0, 1] is an ambient isotopy of M.

Theorem 1.12.2. (Minimax Theorem) Suppose that M is a smooth man-ifold with a complete Finsler metric. Suppose, moreover, that f :M→ [0,∞)is a C2 function satisfying condition C and F is a nonempty family of subsetsof M which is ambient isotopy invariant. Then

Minimax(f,F) = inf supf(p) : p ∈ A : A ∈ F

is a critical value for f .

Proof: Let c = Minimax(f,F). If c is not a critical value for f , then ConditionC implies that there exists an ε > 0 such that there are no critical points p ∈Mwith f(p) ∈ (c − ε, c + ε). But by definition of Minimax(f,F), there exists anA ∈ F such that A ⊂ Mc+ε. Theorem 1.11.1 then gives a smooth isotopyψt : t ∈ [0, 1] such that

ψ1(Mc+ε) ⊂Mc−ε, so ψ1(A) ⊂Mc−ε.

But then ψ1(A) ∈ F showing that Minimax(f,F) ≤ c− ε, a contradiction.

For example, we could let M0 be a component of M and let

F = A ⊆M : A ⊆M0.

Then F is ambient isotopy invariant and the previous Theorem implies:

Corollary 1.12.3. Suppose that M is a smooth manifold with a completeFinsler metric, and that f : M → [0,∞) is a C2 function satisfying conditionC. Then f assumes its minimum value on each component of M.

We can also use invariants from algebraic topology to construct critical points.For example, suppose that M is simply connected and [α] is a nonzero elementin πk(M), the k-th homotopy group of M. In this case,

F[α] = h(Sk) such that h : Sk →M is a continuous map representing [α]

is ambient isotopy invariant, and hence there is a minimax critical point corre-sponding to the homotopy class [α]. Alternatively, suppose that x is a nonzeroelement in Hk(M; R), the singular homology group of M of degree k with realcoefficients, and let

Fx = h(A) such that A is a compact oriented manifold of dimension k

with fundamental class [µA] and h : A→M is acontinuous map with h∗([µA]) a nonzero multiple of x.

Once again Fx is ambient isotopy invariant, and one obtains a minimax criticalpoint corresponding to the homology class x. For a third example, we suppose

55

that θ is a differential k-form on M such that dθ = 0, and let

Fθ = h(A) such that A is a compact oriented manifold of dimension k

and h : A→M is a C1-map such that∫A

h∗θ 6= 0.

It follows from the Homotopy Lemma to be proven in the next section that Fθis ambient isotopy invariant, so once again we obtain a minimax critical pointcorresponding to the differential form θ.

Theorem 1.12.4. Suppose that K is the set of critical points for f whosecritical value is Minimax(f,F) and that U is an open neighborhood of K withinM. If the elements in F are compact, then there is an element A ∈ F such thatfor some ε > 0,

φt(A) ⊂Mc−ε ∪ U, for all t ≥ 0.

Proof: We follow an argument presented by Klingenberg [42]. First choose anopen subset V ofM such that K ⊂ V ⊂ U and the distance from V toM−U isa positive number δ. It follows from condition C that the norm of df is boundedbelow on M− V—otherwise, a sequence of points in M− V would convergeto a critical point which would not lie in K. Moreover, any orbit of −X whichenters V can only leave U if it travels for a distance δ in U − V , but then itfollows from (1.21) that the value of f must decrease by at least δ. Thus if weset ε = δ/2, there will exist an element A ∈ F such that A ⊂Mc+ε, and whenT is sufficiently large, every orbit starting in A will have either passed belowlevel c−ε or entered V at least once in time T . Hence A′ = φT (A) is an elementof F and φt(A′) ⊂Mc−ε ∪ U for all t ≥ 0.

1.13 de Rham cohomology

Once one has C2 partitions of unity on Banach manifolds, it is relatively straight-forward to extend de Rham cohomology to Banach manifolds. Indeed, C2 par-titions of unity will enable us to piece together C1 differential forms with C1

exterior derivatives, elements of the vector space

Ωk(M) = C1 differential forms ω on M of degree k : dω ∈ C1.

We make the Ωk(M)’s into a cochain complex in which the differential is theexterior derivative

d : Ωk(M) −→ Ωk+1(M).

We say that an element ω ∈ Ωk(M) is closed if dω = 0 and exact if ω = dθfor some θ ∈ Ωk−1(M). Since d d = 0, every exact k-form is closed. Thequotient space

HkdR(M; R) =

closed elements of Ωk(M)exact elements of Ωk(M)

.

56

is called the de Rham cohomology of M. If ω ∈ Ωk(M) is closed, we let [ω]denote its cohomology class in Hk

dR(M; R). In the terminology of algebraictopology, the de Rham cohomology is the cohomology of the cochain complex

· · · → Ωk−1(M)→ Ωk(M)→ Ωk+1(M)→ · · · . (1.23)

Note that de Rham cohomology has a cup product defined by

[ω] ∪ [φ] = [ω ∧ φ],

which makes the direct sum

H∗dR(M; R) =∞∑k=0

HkdR(M; R)

into a graded commutative algebra over the ring of smooth real-valued functionson M . Moreover, the cup product behaves well under smooth maps: If F :M→N is a smooth map, the linear map F ∗ on differential forms induces a linearmap

F ∗ : HkdR(N ; R) −→ Hk

dR(M; R) such that F ∗([ω] ∪ [φ]) = F ∗[ω] ∪ F ∗[φ].

Moreover, the identity map onM induces the identity on de Rham cohomologyand if F :M→N and G : N → P are smooth maps, then (G F )∗ = F ∗ G∗,so

M 7→ Hk(M; R), (F :M→N ) 7→ (F ∗ : HkdR(N ; R)→ Hk

dR(M; R))

is a contravariant functor from the category of smooth manifolds and smoothmaps to the category of real vector spaces and linear maps.

Lemma 1.13.1. (Poincare Lemma.) If U is a convex open subset of aBanach space E, or more generally any contractible open subset of E, then thede Rham cohomology of U is trivial:

HkdR(U ; R) ∼=

R if k = 0,

0 if k 6= 0.

One can modify the proof that is used in the finite-dimensional case. We onlysketch the key ideas—the reader should refer to Lang [43] for details. Since theinclusion from a point into the convex set U is a homotopy equivalence, thePoincare Lemma is an immediate consequence of

Lemma 1.13.2. (Homotopy Lemma.) Smoothly homotopic maps F,G :M→N induce the same map on cohomology,

F ∗ = G∗ : HkdR(N ; R) −→ Hk

dR(M; R).

57

On the other hand via functoriality, this follows from the special case of theHomotopy Lemma for the inclusion maps

i0, i1 :M−→ [0, 1]×M, i0(p) = (0, p), i1(p) = (1, p).

Indeed, if H : [0, 1] ×M → N is a smooth homotopy from F to G, then bydefinition of homotopy, F = H i0 and G = H i1, so

i∗0 = i∗1 ⇒ F ∗ = i∗0 H∗ = i∗1 H∗ = G∗.

This special case, however, can be established by integrating over the fiberof the projection on the second factor [0, 1] ×M → M. More precisely, let tbe the standard coordinate on [0, 1], T the vector field tangent to the fiber of[0, 1]×M such that dt(T ) = 1. We then define integration over the fiber

π∗ : Ωk([0, 1]×M)→ Ωk−1(M) by π∗(ω)(p) =∫ 1

0

(ιTω)(t, p)dt.

Here the interior product (ιTω)(t, p) is an element of ΛkT ∗(t,p)([0, 1] ×M) andthe integration is possible because the exterior power at (t, p) is canonically iso-morphic to ΛkT ∗(0,p)([0, 1]×M). The key to proving that i∗0 = i∗1 in cohomologyis the “cochain homotopy” formula

i∗1ω − i∗0ω = d(π∗(ω)) + π∗(dω).

This formula can be verified by using naturality to reduce the proof to thefinite-dimensional case, and then calculating in local coordinates just as in thefamiliar finite-dimensional treatment found in [10]. Note that

dω = 0⇒ i∗1ω − i∗0ω = d(π∗(ω))⇒ [i∗1ω] = [i∗0ω],

and hence on the cohomology level i∗0 = i∗1. This finishes our sketch of the proofof the Homotopy Lemma and the Poincare Lemma.

Remark 1.13.3. If N is a submanifold of M with inclusion map i : N →M,we let

Ωk(M,N ) = ker(i∗ : Ωk(M)→ Ωk(N )),

and note that the exterior derivative makes this into the k-th cochain group of acochain complex Ω∗(M,N ). The cohomology of this complex is called the rela-tive de Rham cohomology of the pair (M,N ) and is denoted by Hk

dR(M,N ; R).The short exact sequence of cochain complexes

0→ Ω∗(M,N )→ Ω∗(M)→ Ω∗(N )→ 0. (1.24)

yields a long exact sequence via the “snake lemma” (Theorem 2.16 in [35]) fromalgebraic topology:

· · · → HkdR(M,N ; R)→ Hk

dR(M; R)→ HkdR(N ; R)

→ Hk+1dR (M,N ; R)→ Hk+1

dR (M; R)→ · · · .

58

This is very useful for calculating de Rham cohomology.For us, one of the primary uses of differential forms will be to calculate

cohomology of infinite-dimensional manifolds. It is important to realize thatthe de Rham cohomology is the same as the singular or Cech cohomology withreal coefficients that is studied in algebraic topology:

Theorem 1.13.4. (de Rham Theorem.) Suppose that eitherM = Lpk(Σ,M)where pk > dim(Σ) or M is finite-dimensional. Suppose, moreover that Madmits C2 partitions of unity. Then the cohomology of the cochain complexΩ∗(M) is isomorphic to the Cech cohomology of M.

The proof (due to Andre Weil) is via the zig-zag construction described in theexcellent text on de Rham theory by Bott and Tu [10], which we follow closely.

In the case where M is finite-dimensional, we let U = Uα : α ∈ A bea locally finite open cover of M by sets which are geodesically convex withrespect to a Riemannian metric on M. If M = Lpk(Σ,M) where pk > dim(Σ),we construct an open cover U = Uα : α ∈ A in which each open set Uα is ofthe form

Uα = g ∈ Lpk(Σ,M) : ‖g − f‖C0 < δ.

The Sobolev Lemma guarantees the existence of such an open cover. We chooseδ so small that if p, q ∈ M and d(p, q) < 2δ, then there is a unique minimizinggeodesic

γp,q : [0, 1]→M such that γp,q(0) = p, γp,q(1) = q,

and moreover this geodesic depends smoothly on p and q. (Then any two pointsin a δ-ball about a given point can be connected by a unique such geodesic.) Ifg1 and g2 are two elements of Uα, we can then define a path

Γg1,g2 : [0, 1]→ Lpk(Σ,M) by Γg1,g2(t)(p) = γg1(p),g2(p)(t).

It is easily checked in either case that the intersection of any collection of el-ements from U is contractible. Readers familiar with cohomology theory willremember that the Cech cohomology of such an open covering is the same asthe Cech cohomology ofM (and we do not need to take direct limits). Such anopen cover is often called a “good” cover or “Leray” cover.

We now construct a double complex K∗,∗ in which the (p, q)-element is

Kp,q = Cp(U ,Ωq),

which is defined to be the space of functions ω which assign to each distinctordered (p + 1)-tuple (α0, . . . , αp) of indices such that Uα0 ∩ · · · ∩ Uαp 6= 0 anelement

ωα0···αp ∈ Ωq(Uα0 ∩ · · · ∩ Uαp)

in such a way that if the order of elements in a sequence is permuted, ωα0,...,αp

changes by the change of the permutation; thus

ωα0α1 = −ωα1α0 , ωαα = 0, and so forth.

59

We have two differentials on the double complex, the exterior derivative

d : Cp(U ,Ωq)→ C

p(U ,Ωq+1) defined by (dω)α0···αp = dωα0···αp ,

and the Cech differential

δ : Cp(U ,Ωq)→ C

p+1(U ,Ωq)

defined by

(δω)α0···αp+1 =p+1∑i=0

(−1)iωα0···αi···αp+1 ,

the forms on the right being restricted to the intersection.The first differential is exact except when q = 0 by the Poincare Lemma,

while in the case q = 0 we find that

[Kernel of d : Cp(U ,Ω0)→ C

p(U ,Ω1)] = C

p(U ; R),

the space of Cech cocycles for the covering U on M. The Cech cohomology ofthe cover U is by definition the cohomology of the cochain complex

· · · → Cp−1

(U ; R)→ Cp(U ; R)→ C

p+1(U ; R)→ · · · (1.25)

and is denoted by Hp(M; R).

The second differential is exact except when p = 0 and it is at this pointthat the C2 partition of unity ψα : α ∈ A subordinate to U is used. Indeed,given a δ-cocycle ω ∈ C

p(U ,Ωq), we set

τα0···αp−1 =∑α

ψαωαα0···αp−1 ∈ Cp−1

(U ,Ωq),

noting that since ψα is C2 we stay in the class of C1 forms with C1 exteriorderivatives. Then

(δτ)α0···αp =∑i,α

(−1)iψαωαα0···αi···αp

and it follows from the fact that δω = 0 thatp∑i=1

(−1)iωαα0···αi···αp = ωα0···αp .

Hence

(δτ)α0···αp =

(∑α

ψα

)ωα0···αp = ωα0···αp ,

establishing exactness. When p = 0, we find that

Kernel of δ : C0(U ,Ωq)→ C

1(U ,Ωq) = Ωq(M),

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the space of smooth q-forms on M.We can summarize the previous discussion by stating that the rows and

columns in the following commutative diagram are exact:

· · ·↑ ↑ ↑

0 → Ω2(M) → C0(U ,Ω2) → C

1(U ,Ω2) → C

2(U ,Ω2) → · · ·

↑ ↑ ↑0 → Ω1(M) → C

0(U ,Ω1) → C

1(U ,Ω1) → C

2(U ,Ω1) → · · ·

↑ ↑ ↑0 → Ω0(M) → C

0(U ,Ω0) → C

1(U ,Ω0) → C

2(U ,Ω0) → · · ·

↑ ↑ ↑C

0(U ,R) C

1(U ,R) C

2(U ,R)

↑ ↑ ↑0 0 0

The remainder of the proof uses this diagram. Given a de Rham cohomol-ogy class [ω] ∈ Hp

dR(M; R) with p-form representative ω we construct a cor-responding cohomology class s([ω]) in the Cech cohomology H

p(M; R) as fol-

lows: The differential form defines an element ω0p ∈ C0(U ,Ωp) by simply re-

stricting ω to the sets in the cover. It is readily checked that ω0p is closedwith respect to the total differential D = δ + (−1)pd on the double com-plex K∗,∗ = C

∗(U ,Ω∗). Using the Poincare Lemma, we construct an element

ω0,p−1 ∈ C0(U ,Ωp−1) such that dω0,p−1 = ω0p. Let ω1,p−1 = δω0,p−1 and ob-

serve that dω1,p−1 = 0 and ω1,p−1 is cohomologous to ω0p with respect to D.Using the Poincare Lemma again, we construct an element ω1,p−2 ∈ C

1(U ,Ωp−2)

such that dω1,p−2 = ω1,p−1. Let ω2,p−2 = δω1,p−2 and note that ω2,p−2 is co-homologous to ω0p with respect to D. Continue in this fashion until we reacha D-cocycle ωp0 ∈ C

p(U ,Ω0) which is cohomologous to ω0p. Since dωp0 = 0,

each function ωp0α0···αp is constant, and thus ωp0 determines a Cech cocycle s(ω)whose cohomology class is s([ω]).

By the usual diagram chasing, the cohomology class obtained is independentof choices made. Moreover, reversing the zig-zag construction described in thepreceding paragraph yields an inverse to s. This finishes our sketch of the proofof de Rham’s theorem; for more details, one can consult [10], Chapter 2.

Remark 1.13.5. The proof shows that the cohomologies of the two cochaincomplexes (1.23) and (1.25) are isomorphic. It follows that the cohomology ofthe cochain complex (1.25) is independent of the choice of good cover. On theother hand, in the case where M has C∞ partitions of unity the argument canbe repeated with C∞ differential forms to show that the de Rham cohomologyis the same whether calculated with C∞ forms of C1 forms with C1 exteriorderivatives.

Remark 1.13.6. This remark assumes some familiarity with singular cohomol-

61

ogy, as treated in Chapter 3 of [35]. One could replace the double complex theoccurs in the proof by

Kp,qs = Cps (U ,Ωq),

which is defined to be the space of functions ω which assign to each distinctordered (p + 1)-tuple (α0, . . . , αp) of indices such that Uα0 ∩ · · · ∩ Uαp 6= 0 anelement sα0···αp in the space of singular cochains within Uα0 ∩ · · · ∩ Uαp withcoefficients in R. The above proof can then be modified to give an isomorphismfrom singular cohomology to the Cech cohomology of M. The argument canalso be modified so that it applies to relative cohomology. Thus readers familiarwith standard cohomology theory can rest assured that de Rham cohomologygives exactly the same results as the cohomology they have studied in algebraictopology courses.

Remark 1.13.7. Suppose we take an arbitrary cover of M , not necessarily agood cover. Then the rows in the above diagram are still exact, even thoughthe columns are not. For example, we can take two open sets U and V suchthat M = U ∪ V . Then the above diagram collapses to a short exact sequenceof de Rham complexes

0→ Ω∗(M)→ Ω∗(U)⊕ Ω∗(V )→ Ω∗(U ∩ V )→ 0.

By the “snake lemma” from algebraic topology, we get a long exact sequence

· · · → HkdR(M ; R)→ Hk

dR(U ; R)⊕HkdR(V ; R)→ Hk

dR(U ∩ V ; R)

→ Hk+1dR (M ; R)→ Hk+1

dR (U ; R)⊕Hk+1dR (V ; R)→ · · ·

which is called the Mayer-Vietoris sequence. The Mayer-Vietoris sequence,together with the Homotopy Lemma, is very helpful in computing de Rhamcohomology.

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Chapter 2

Morse Theory of Geodesics

2.1 Geodesics

Our next goal is to explain how critical point theory on infinite-dimensionalmanifolds can be used to used to produce periodic solutions to a class of im-portant nonlinear ordinary differential equations, the equations for geodesics inRiemannian manifolds.

The geodesic equation is a generalization of the simplest second-order lin-ear ordinary differential equation—the equation of a particle moving with zeroacceleration in Euclidean space. This asks for a vector-valued function

γ : (a, b) −→ RN such that γ′′(t) = 0,

and its solutions are the constant speed straight lines. The simplest way tomake this differential equation nonlinear is to consider a proper submanifold Mof RN with the induced Riemannian metric, and ask for a function

γ : (a, b) −→M ⊂ RN such that (γ′′(t))T = 0,

where (·)T denotes projection into the tangent space of M . In simple terms,we are asking for the curves which are as straight as possible subject to theconstraint that they lie within M . In the terminology of differential geometry,the tangential projection of the ordinary derivative is known as the covariantderivative, and one often writes

(γ′′(t))T = ∇γ′γ′(t) or (γ′′(t))T =Dγ′

dt(t),

where ∇ and D are two commonly used notations for the covariant derivative.The smooth maps γ which satisfy the equation ∇γ′γ′(t) = 0 are called thesmooth geodesics in M .

We can put the geodesic equation into the more general context of simplemechanical systems: We let (M, 〈·, ·〉) be a Riemannian manifold, which we can

63

assume has a proper isometric imbedding ι : Mn → RN into Euclidean space;such an imbedding is provided by the Nash imbedding theorem. In addition, weconsider a smooth function φ : M → R, to be called the potential . The triple(M, 〈·, ·〉, φ) is called a simple mechanical system. For a simple mechanicalsystem, “Newton’s equation of motion” is

∇γ′γ′(t) = −(grad φ)(γ′(t)).

The left-hand side can be interpreted as the acceleration of a moving particleof unit mass, while the right-hand side is the force (per unit mass) produced bythe potential φ.

As we mentioned before, the idea behind the calculus of variations is toregard solutions to ordinary differential equations—such as the geodesic equa-tion or Newton’s equation of motion—as critical points of functions defined oninfinite-dimensional manifolds. In the case of geodesics, the infinite-dimensionalmanifold is

L21([0, 1],M) = γ ∈ L2

1([0, 1],RN ) : γ(t) ∈M , for all t ∈ [0, 1] ,

where M is a proper submanifold of RN , or one of the many useful subspacesof L2

1([0, 1],M). These include the free loop space

L21(S1,M) = γ ∈ L2

1([0, 1],M) : γ(0) = γ(1),

the space of paths from p to q,

Ω(M,p, q) = γ ∈ L21([0, 1],M) : γ(0) = p, γ(1) = q,

where p and q are points of M , and the space of paths from S1 to S2,

Ω(M,S0, S1) = γ ∈ L21([0, 1],M) : γ(0) ∈ S0, γ(1) ∈ S1,

when S0 and S1 are compact imbedded submanifolds of M .The first step towards formulating the equations of geodesics within critical

point theory is to define the Euclidean action

JRN : L21([0, 1],RN )→ R by JRN (γ) =

12

∫ 1

0

γ′(t) · γ′(t)dt,

the dot denoting the Euclidean dot product. The map JRN is clearly smooth,being the restriction of a continuous bilinear map

(γ, λ) 7→ 12

∫S1γ′(t) · λ′(t)dt

to the diagonal.Recall from §1.4 that the isometric imbedding ι : M → RN induces by

composition a map

ωι : L21([0, 1],M)→ L2

1([0, 1],RN )

64

which is also a smooth imbedding. Moreover, the Hilbert space structure onthe spaces L2

1([0, 1],RN ) induces Riemannian metrics on L21([0, 1],M) and its

subspaces L21(S1,M) and Ω(M,p, q). We define

JM : L21([0, 1],M)→ R by JM = JRN ωι,

a map which is clearly smooth, being the composition of smooth maps. Inaddition, if φ : M → R is a smooth function, the map

γ ∈ L21([0, 1],M) 7→

∫ 1

0

φ(γ(t))dt

is also smooth, since it is the composition of

ωφ : L21([0, 1],M)→ L2

1([0, 1],R)

with integration, integration being a continuous linear map. Finally, we definethe action for the simple mechanical system (M, 〈·, ·〉, φ) to be the function

JM,φ : L21([0, 1],M)→ R, JM,φ(γ) =

∫ 1

0

[12〈γ′(t), γ′(t)〉 − φ(γ(t))

]dt.

By restriction, we also get smooth maps

JM,φ : L21(S1,M)→ R, JM,φ : Ω(M,p, q)→ R.

With these preparations out of the way, we can now state Hamilton’s prin-ciple of least action: the motion of a simple mechanical system is described bya critical point for the action function JM,φ on L2

1(S1,M) or Ω(M,p, q).Focusing first on the case of free loops, the case needed for studying periodic

motion, we are led to ask: what is the differential

(dJM,φ)γ : Tγ(L21(S1,M)) −→ R?

We will assume that γ is smooth and that V ∈ Tγ(L21(S1,M)) lies in the space

of C∞ sections of γ∗TM .To calculate the differential, we suppose that α : S1 × (−ε, ε) → M is a

smooth one parameter family of curves with α(t, 0) = γ(t) and D2α(t, 0) = V (t),where D2 denote the partial derivative with respect to the second slot. Then

ωα : L21(S1, S1 × (−ε, ε)) −→ L2

1(S1,M)

is smooth. Let

µ : (−ε, ε)→ L21(S1, S1 × (−ε, ε)) by µ(τ)(t) = (t, τ),

for t ∈ S1. Clearly, µ is smooth and hence α = ωα µ is a smooth curve inL2

1(S1,M) with α′(0) = X. A straightforward calculation now shows that

dJM,φ(γ)(V ) =d

dt(JM,φ α)

∣∣∣∣τ=0

=∫S1

[〈γ′(t),∇γ′V 〉 − dφ(γ(t))(V (t))]dt,

65

where ∇γ′V is the directional derivative of V in the direction of γ′ projectedinto the tangent space, otherwise known as the covariant derivative of V . Sinceγ is assumed to be smooth, we can integrate by parts to obtain

dJM,φ(γ)(V ) = −∫S1〈∇γ′γ′(t) + (grad φ)(γ′(t)), V (t)〉dt, (2.1)

the boundary terms cancelling by periodicity. Since X can be an arbitrarysmooth vector field along γ, a critical point for JM,φ must be a periodic solutionto Newton’s equation of motion,

∇γ′γ′(t) = −(grad φ)(γ′(t)). (2.2)

Thus Hamilton’s principle implies that the motion of a simple mechanical systemshould be represented by solutions to Newton’s equation of motion. In the casewhere φ = 0, a critical point for the action is simply a smooth closed geodesic.

In a quite similar fashion, one finds that critical points to JM,φ : Ω(M ; p, q)→R are solutions γ to (2.2) such that γ(0) = p and γ(1) = q. On the other hand,if one calculates the derivative on the larger space Ω(M,S0, S1), the integrationby parts gives additional boundary terms and we must replace (2.1) by

dJM,φ(γ)(V ) = −∫S1〈∇γ′γ′(t) + (grad φ)(γ′(t)), V (t)〉dt

+ 〈γ′(1), V (1)〉 − 〈γ′(0), V (0)〉.

This more complicated formula must be used when considering critical pointsfor dJM,φ : Ω(M,S0, S1) → R, and in this case V (0) and V (1) are constrainedto be tangent to S0 and S1. The first variation formula implies that criticalpoints are solutions to (2.2) which are perpendicular to S0 and S1.

2.2 Condition C for the action

In order to apply the method of steepest descent to calculus of the variationsproblems described in the preceding section, we need the topology on the spaceof maps to satisfy two conditions:

1. It must be strong enough so that JM,φ is continuous.

2. It must be weak enough so that suitable sequences γi (such as sequencessuch that JM,φ(γi) tends to a minimum or minimax value subject to someconstraint) will converge in the topology.

If we chose the topology to be too strong, this would make it difficult to es-tablish convergence of γi. The two conflicting requirements single out L2

1 asthe appropriate space to use when studying critical points of the action JM,φ.indeed, we next show that when M is compact, the action functions

JM , JM,φ : L21(S1,M) −→ R

66

satisfy condition C, thereby making available the minimax principle from §1.11.Moreover, we will show that the critical points of J are actually C∞ curves.

Theorem 2.2.1. If (M, 〈·, ·〉) is a compact Riemannian manifold, the functionJM : L2

1(S1,M)→ R, defined by

JM (γ) =12

∫S1〈γ′(t), γ′(t)〉dt,

satisfies Condition C: if γi is a sequence in L21(S1,M) such that

JM (γi) is bounded and ‖dJM (γi)‖ → 0, (2.3)

then it possesses a subsequence which converges to a critical point for JM .

Proof: We start by recalling the proof of the Sobolev Lemma from §1.4. Supposethat γi is a sequence of elements of L2

1(S1,M) satisfying (2.3). We can regardeach γi as an element of the space L2

1(S1,RN ). Then for t1 < t2,

|γi(t1)− γi(t2)| ≤∫ t2

t1

|γ′i(t)|dt ≤√t2 − t1

[∫ t2

t1

|γ′i(t)|2dt]1/2

≤√t2 − t1

√2JRN (γi).

Since JM (γi) is bounded, we see that γi is equicontinuous. Since γi takesvalues in the compact submanifold M ⊂ RN , γi is also uniformly bounded.It therefore follows from Arzela’s theorem or Ascoli’s theorem ([66], page 179)that a subsequence of γi will converge uniformly to a continuous map γ∞ :S1 → M . For simplicity, we continue to denote the subsequence by γi. Tofinish the proof, we need to show that

‖dJM (γi)‖ → 0 ⇒ γi is a Cauchy sequence in L21(S1,M).

To understand |dJM (γ)‖ we need to be able to compare an element ofTγL

21(S1,RN ) with its projection into TγL

21(S1,M). Recalling that M is a

submanifold of RN with inclusion i : M → RN , we let P denote the vectorbundle map

P : i∗T (RN ) −→ TM

which projects onto the tangential component. By the ω-Lemma, we have asmooth map

ωP : L21(S1, i∗T (RN )) −→ L2

1(S1, TM)

which is also a vector bundle map.

Lemma 2.2.2. If 〈·, ·〉 denotes the Riemannian metric on either L21(S1,RN ) or

L21(S1,M) and X ∈ L2

1(S1, i∗T (RN )) with ωπ(X) = γ, then

〈ωP (X), ωP (X)〉 ≤ [1 + CJM (γ)]〈X,X〉,

where C is some constant which depends on M but is independent of γ.

67

Proof: We can regard P as a section of the bundle i∗(End(RN )) over M and itpossesses a differential dP which is a section of T ∗M ⊗ [i∗(End(RN ))]. Let

C1 = sup‖dP (v)‖ : v is a unit-length element of TM ,

a finite constant since M is compact. Then it follows from the Leibniz rule that

(ωP (X))(t) = Pγ(t)X(t) ⇒ (ωP (X))′(t) = dP (γ′(t))(X(t)) + Pγ(t)X′(t).

Recall that

〈ωP (X), ωP (X)〉 =∫S1

[‖(ωP (X))′(t)‖2 + ‖(ωP (X))(t)‖2

]dt,

where ‖ · ‖ denotes the norm in RN , and hence

〈ωP (X), ωP (X)〉 =∫S1

[‖dP (γ′(t))(X(t)) + Pγ(t)X′(t)‖2 + ‖Pγ(t)(X)(t)‖2]dt

≤∫S1

[‖dP (γ′(t))(X(t))‖2 + ‖Pγ(t)X′(t)‖2 + ‖Pγ(t)(X))(t)‖2]dt

≤∫S1

[C1‖γ′(t)‖2‖(X(t)‖2 + ‖X ′(t)‖2 + ‖X(t)‖2]dt.

Thus

〈ωP (X), ωP (X)〉 ≤ C1 sup‖(X(t)‖2 : t ∈ S1∫S1‖γ′(t)‖2dt+ 〈X,X〉

≤ (CJM (γ) + 1)〈X,X〉,

for some positive constant C. This finishes the proof of the lemma.

Lemma 2.2.3. If X ∈ L21(S1, i∗T (RN )) and ωπ(X) = γ, where

ωπ : L21(S1, i∗T (RN ))→ L2

1(S1,M)

is the projection induced by π : i∗T (RN )→M , then

dJM (γ)(ωP (X)) =∫S1

[γ′(t) ·X ′(t) + α(γ′(t), γ′(t)) ·X(t)]dt, (2.4)

where α : TM × TM → NM is the second fundamental form of M in RN andthe dot products are taken in the ambient Euclidean space RN .

Proof: Note that it suffices to establish (2.4) in the case where γ and X aresmooth because both sides are continuous in the L2

1-topology. It follows from(2.1) with φ = 0 that

dJM (γ)(ωP (X)) = −∫S1γ′′(t) · Pγ(t)(X(t))dt.

68

But since α(γ′(t), γ′(t)) is the normal component of γ′′(t),

γ′′(t) ·X(t) = Pγ(t)(γ′′(t)) ·X(t) + α(γ′(t), γ′(t)) ·X(t)= γ′′(t) · Pγ(t)(X(t)) + α(γ′(t), γ′(t)) ·X(t),

and hence

dJM (γ)(ωP (X)) =∫S1

[−γ′′(t) ·X(t) + α(γ′(t), γ′(t)) ·X(t)]dt.

An integration by parts yields the claim.

Returning to the proof of the theorem, we let γi be a sequence satisfying (2.3)which converges uniformly to a continuous map γ∞ : S1 →M . Since JM (γi) isbounded, 〈γi, γi〉 = ‖γi‖2 is bounded, and hence

‖γi − γj‖2 ≤ (‖γi‖+ ‖γj‖)2

is also bounded as i, j → ∞. Lemma 2.2.2 implies that ‖(ωP )γi(γi − γj)‖ isbounded as well. Since ‖dJM (γi)‖ → 0, for every ε > 0 there exists N ∈ N suchthat

|dJM (γi)((ωP )γi(γi − γj))− dJM (γj)((ωP )γj (γi − γj))| < ε (2.5)

for i, j > N . Here (ωP )γi(γi − γj) lies in the tangent space to L21(S1,M) at γi.

It follows from (2.5) and the explicit formula (2.4) for dJM that for every ε > 0there exists N ∈ N such that∫

S1[γ′i(t) · (γ′i(t)− γ′j(t)) + α(γ′i(t), γ

′i(t)) · (γi(t)− γj(t))]dt

−∫S1

[γ′j(t) · (γ′i(t)− γ′j(t)) + α(γ′j(t), γ′j(t)) · (γi(t)− γj(t))]dt < ε,

for i, j > N . But|α(γ′i(t), γ

′i(t))| ≤ (constant)|γ′i(t)|2,

the constant is a bound for the norm of the second fundamental form of M andhence depends only on the submanifold M . Since |γi(t)− γj(t)| < 1/n for i andj sufficiently large,∣∣∣∣∫

S1α(γ′i(t), γ

′i(t)) · (γi(t)− γj(t))dt

∣∣∣∣ ≤ (constant)JM (γi)1n→ 0

as i, j → ∞, a similar implication holding when i is replaced by j. Hence forevery ε > 0 there exists N ∈ N such that∣∣∣∣∫

S1(γ′i(t)− γ′j(t)) · (γ′i(t)− γ′j(t))dt

∣∣∣∣ < ε,

for i, j > N . This, together with the C0 convergence of γi implies that γiis a Cauchy sequence in L2

1(S1,RN ). Since L21(S1,RN ) is complete, γi has a

69

limit in L21(S1,RN ), which must of course be γ∞. Thus γi → γ∞ ∈ L2

1(S1,M),and by continuity of dJM , γ∞ must be a critical point for JM . This concludesthe proof of Theorem 2.2.1.

Theorem 2.2.4. Suppose that (M, 〈·, ·〉) is a compact Riemannian manifold,φ : M → R is a smooth function and ψ : S1 → RN is an L2

1 map. The real-valuedfunctions JM,φ and JM,ψ on L2

1(S1,M), defined by

JM,φ(γ) =12

∫S1

[〈γ′(t), γ′(t)〉 − φ(γ(t))] dt and

JM,ψ(γ) =12

∫S1

[〈γ′(t), γ′(t)〉+ γ(t) · ψ(t)] dt,

satisfy Condition C.

Sketch of proof: Suppose that γi is a sequence in L21(S1,M) such that JM,φ(γi)

is bounded and ‖dJM,φ(γi)‖ → 0. Then JM (γi) is also bounded and just asbefore, γi is uniformly bounded and equicontinuous and thus by the Arzela-Ascoli theorem, possesses a subsequence which converges uniformly to a contin-uous map γ∞.

We can now mimic the preceding proof up to equation (2.5). At this point,we need to account for an extra term in dJM,φ, namely

−∫S1dφ(γi(t))(Pγi(t)(γi(t)− γj(t)))dt.

However, this term goes to zero, since γi(t) is Cauchy in C0. Similarly, thereis an extra term in JM,ψ which goes to zero. With these minor changes, theargument proceeds to the desired conclusion exactly as before.

For the statement of the next theorem, we suppose that M is a complete Rie-mannian manifold which has a proper isometric immersion into some Euclideanspace RN .

Theorem 2.2.5. Suppose that (M, 〈·, ·〉) is a compact Riemannian manifoldand φ : M → R is a smooth function. The real-valued functions

JM,φ : Ω(M,p, q)→ R and JM,φ : Ω(M,S1, S2)→ R,

defined by

JM,φ(γ) =12

∫ 1

0

[〈γ′(t), γ′(t)〉 − φ(γ(t))] dt,

satisfy condition C.

Proof: A straightforward modification of preceding arguments.

Theorem 2.2.6. if the path γ within L21(S1,M), Ω(M,p, q) or Ω(M,S1, S2) is

a critical point for JM or JM,φ, then γ is C∞. Moreover, if ψ ∈ L2k(S1,RN ),

then any critical point for JM,ψ lies in L2k+2(S1,M).

70

Proof: We prove only the free loop space cases. If γ is a critical point for JM ,it follows from (2.4) that∫

S1γ′(t) ·X ′(t)dt = −

∫S1α(γ′(t), γ′(t)) ·X(t)dt,

for all X ∈ TγL21(S1,RN ). Thus in the sense of distributions,

γ′′(t) = α(γ′(t), γ′(t)). (2.6)

Note thatγ ∈ L2

1 ⇒ γ′ ∈ L2 ⇒ α(γ′(t), γ′(t)) ∈ L1.

Thus it follows from (2.6) and standard theorems of analysis that γ is C1. Nowwe use the technique of “elliptic bootstrapping”:

γ ∈ C1 ⇒ RHS of (2.6) is C1 ⇒ γ ∈ C2 ⇒ RHS of (2.6) is C2 ⇒ · · · .

By induction, we see that γ is C∞ and Theorem 2 is established for JM .The proofs for JM,φ and JM,ψ are the same except that (2.6) is replaced by

γ′′(t) = α(γ′(t), γ′(t))− (grad φ)(γ′(t))

or γ′′(t) = α(γ′(t), γ′(t)) + (ψ(t))T ,

where (·)T is the tangential component.

2.3 Existence of smooth closed geodesics

In studying closed geodesics on a Riemannian manifold (M, 〈·, ·〉), we let M =L2

1(S1,M), an infinite-dimensional Hilbert manifold with a complete Rieman-nian metric, and define the action

J :M−→ R by J(γ) =12

∫S1〈γ′(t), γ′(t)〉dt,

a function which satisfies condition C by Theorem 2.2.1. We have seen thatM = L2

1(S1,M) is homotopy equivalent to C0(S1,M), the free loop spacestudied by topologists.

Theorem 2.3.1. Let (M, 〈·, ·〉) be a compact connected Riemannian manifoldand let [S1,M ] denote the set of free homotopy classes of continuous maps fromS1 to M . If α ∈ [S1,M ] and

L21,α(S1,M) = γ ∈ L2

1(S1,M) : the free homotopy class of γ is α,

then J assumes its minimum on L21,α(S1,M). If α 6= 0, this minimum is achieved

at a nonconstant smooth closed geodesic.

Proof: This is a direct consequence of Theorem 2.2.1 and Corollary 1.12.3.

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The preceding theorem shows a nonsimply connected compact manifold alwayspossesses a smooth closed geodesic. To treat the simply connected case, we needthe following theorem due to Lusternik and Fet:

Theorem 2.3.2. If (M, 〈·, ·〉) is a compact simply connected Riemannian man-ifold, M contains a nonconstant smooth closed geodesic.

Before proving this theorem, we recall some concepts we need from homotopytheory. A fibration is a continuous map f : E → B which has the homotopylifting property; this means that if the continuous map g : Y → E has theproperty that its projection into the base f g : Y → M can be extended to ahomotopy

H : Y × [0, 1]→ B with H(y, 0) = f g(y), for y ∈ In,

then this homotopy H can be lifted to

H : Y × [0, 1]→ E such that H(y, 0) = g(y) and f H = H.

The key facts about fibrations are that the fibers Ep = f−1(p), for p ∈ B, arehomotopy equivalent to each other, and the map f induces a long exact sequenceof homotopy groups

· · · → πk(Ep)→ πk(E)→ πk(B)→ πk−1(Ep)→ · · · .

The reader can refer to [10], §§16 and 17 or [35], Theorem 4.41 for proofs ofthese and related facts. In fact, the only thing needed for the exact sequenceis that f be a weak fibration, which means that it has the homotopy liftingproperty for the case where Y = [0, 1]n, the n-cube, for all choices of n.

A key example concerns the path space

P = continuous maps γ : [0, 1]→M such that γ(0) = p ,

where p is some choice of base point in M . In this case, the map

π : P →M, π(γ) = γ(1)

is a fibration. Indeed, given

g : Y → P and H : Y × [0, 1]→M with H(y, 0) = π g(y),

we can define the lift H : Y × [0, 1]→ P by

H(y, s)(t) =

g(y)

(t

1−(s/2)

), for 0 ≤ t ≤ 1− (s/2),

H(y, 2t− 2 + s), for 1− (s/2) ≤ t ≤ 1.

One readily checks that H is continuous and has the desired properties. Indeed,when t = 1− (s/2),

g(y)(

t

1− (s/2)

)= g(y)(1) = H(y, 0) = H(y, 2t− 2 + s),

72

so the two pieces of the function fit together continuously, while when we sett = 1, we find that H(y, s)(1) = H(y, 2− 2 + s) = H(y, s), so H is indeed a liftH.

The fiber over the base point p of this fibration is

Ωp = continuous maps γ : [0, 1]→M such that γ(0) = p = γ(1)

and is known as the pointed loop space. Its homotopy groups can be computedby the long exact sequence of the fibration,

· · · → πk(Ωp)→ πk(P)→ πk(M)→ πk−1(Ωp)→ · · · .

Since P is contractible via the homotopy

H : P × [0, 1]→ P, where H(γ, s)(t) = γ((1− s)t),

we see that πk(P) = 0 for all k and this long exact sequence collapses to yield

πk(Ωp) ∼= πk+1(M).

A second example is the free loop space

C0(S1,M) = continuous maps γ : [0, 1]→M such that γ(0) = γ(1) ,

which is the total space of a fibration

ev : C0(S1,M)→M defined by ev(γ) = γ(0).

The fiber over p in this case is Ωp once again, so we obtain a long exact sequence

· · · → πk(Ωp)→ πk(C0(S1,M))→ πk(M)→ πk−1(Ωp)→ · · · .

In this case, however, ev∗ : πk(C0(S1,M))→ πk(M) possesses a right inverse

i∗ : πk(M)→ πk(C0(S1,M)) induced by the map i : M → C0(S1,M)

which takes the point p ∈M to the constant loop located at p. Hence the exactsequence for ev splits, and we conclude that

πk(C0(S1,M)) ∼= πk(M)⊕ πk(Ωp) ∼= πk(M)⊕ πk+1(M).

Thus we see that the homotopy groups of the free loop space C0(S1,M) arequite easily determined from the homotopy groups of M .

Proof of Theorem 2.3.2: Since π1(M) = 0, M is orientable and henceHn(M ; Z) 6=0. Let q be the smallest positive integer such that Hq(M,Z) 6= 0. It followsfrom the Hurewicz isomorphism theorem that

πi(M) = 0, for 0 < i < q, and πq(M) ∼= Hq(M,Z) 6= 0.

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It follows from Theorem 1.5.1 that M = L21(S1,M) is homotopy equivalent to

C0(S1,M). Hence

πk(M) ∼= πk(M)⊕ πk(Mp) ∼= πk(M)⊕ πk+1(M).

In particular,πq−1(M) ∼= πq(M) 6= 0.

Since M is simply connected, q ≥ 2 and πq−1(M) ∼= πq(M) is abelian.Moreover, we can identify πq−1(M) with [Sq−1,M], the space of free homotopyclasses of maps from Sq−1 toM. Choose a nonzero element α ∈ [Sq−1,M]. Let

F = g(Sq−1) such that g : Sq−1 →M is a continuous map in [α].

Clearly, F is an ambient isotopy invariant family of sets. Hence Minimax(J,F)is a critical value for J . We need only verify that Minimax(J,F) ≥ ε for someε > 0. Note that by the Cauchy-Schwarz inequality, J−1([0, ε]) consists of curvesof length ≤

√2ε.

Recall that M is isometrically imbedded in an ambient Euclidean space RN .If p ∈ M , let NpM denote the normal space to M in RN and let νM be thedisjoint union of all the normal spaces NpM , for p ∈ M , the total space of asmooth vector bundle over M , called the normal bundle. Let νM(δ) denote theunion of all the balls of radius δ > 0 in N − pM , for p ∈ M , and let let M(δ)denote the open δ-neighborhood of M in RN . If δ > 0 is sufficiently small, onecan prove that the exponential map

exp : (νM)(δ) −→M(δ),

is a diffeomorphism; this is the content of the tubular neighborhood theoremfrom differential topology ([36], Chapter 4, §5). Thus if δ is sufficiently small,M is a strong deformation retract of M(δ) and hence M is homotopy equivalentto M(δ).

For ε > 0 sufficiently small, any closed curve γ on M such that J(γ) < ε andhence of length <

√2ε (by the Cauchy-Schwarz inequality) can be contracted

to the point γ(0) in RN without leaving M(δ). Thus if g(Sq−1) ⊂ J−1([0, ε]),then g is homotopic to a smooth map

g : Sq−1 →M0, where M0 = γ ∈M : γ is constant .

But πq−1(M0) = πq−1(M) = 0, and hence g is homotopic to a constant, contra-dicting the definition of F . Hence Minimax(J,F) ≥ ε, and M must possess anonconstant smooth closed geodesic.

Remark 2.3.3. The technique we have used to prove the Lusternik-Fet Theo-rem, based upon the minimax principle, is often called Lusternik-Schnirelmanntheory .

74

2.4 Second variation

A defect in the minimax principle is that quite different topological constraintscan in fact lead to the same minimax critical points. We need a more refinedtheory that can analyze the contributions to the topology made by each individ-ual critical point. The contributions of “nondegenerate” critical points are theeasiest to analyze. In this section, we take the first step towards understandingnondegenerate critical points by investigating the structure of the Hessian at acritical point.

Let M be a Banach manifold and let f : M → R be a smooth map. Ifp ∈ M is a critical point for f , then the Hessian of f at p is the symmetricbilinear map d2f(p) : TpM×TpM→ R defined in terms of a chart (Uα, φα) by

d2f(p)([α, p, v], [α, p, w]) = D2(f φ−1α )(φα(p))(v, w).

It is straightforward to show that this is independent of choice of chart. Indeed,if α : (−ε, ε)→M is a smooth curve with α(0) = p and α′(0) = V , then a shortcalculation shows that

d2f(p)(V, V ) =d2

dt2(f α)

∣∣∣∣t=0

,

an expression which is clearly independent of choice of chart. Of course, once oneknows d2f(p)(V, V ), one can determine d2f(p)(V,W ) for arbitrary V,W ∈ TpMby the polarization identity:

d2f(p)(V,W ) =14

[d2f(p)(V +W,V +W )− d2f(p)(V −W,V −W )].

Definition. The Morse index of a critical point p for f is the maximal dimen-sion of a linear subspace of TpM on which d2f(p) restricts to a negative definitesymmetric bilinear form. The nullity of the critical point p is

dim V ∈ TpM such that d2f(p)(V,W ) = 0 for all W ∈ TpM .

We say that the critical point is stable if its Morse index is zero. In some sense,the index measures the extent to which a critical point fails to be stable.

We would like to calculate the Hessian at critical points for the simple me-chanical systems we described in §2.1, where we take M to be a space of L2

1

maps. In the case whereM = L21(S1,M), it is proven in elementary differential

geometry texts that the Hessian of the action J is given by the second variationformula or index formula:

Proposition 2.4.1. If (M, 〈·, ·〉) is a compact Riemannian manifold and J :L2

1(S1,M)→ R is the usual action, then

d2J(γ)(V,W ) =∫S1

[〈∇γ′(t)V,∇γ′(t)W 〉 − 〈R(V, γ′(t))γ′(t),W 〉]dt, (2.7)

75

for V,W ∈ TγM.

Let us review how to establish (2.7). We consider a smooth variation of γ whichhas its support within a given coordinate chart (U, t) on S1. Recall that such avariation is a smooth family of maps u 7→ γu in L2

1(S1,M) and let

α(t, u) = γu(t), V (t) =∂α

∂u(t) ∈ Tγ(t)M.

Differentiating twice, we obtain

d2(J)(γ)(V, V ) =d2

du2(J(γu))

∣∣∣∣u=0

=∫S1

∂

∂u

⟨∂α

∂t,D

∂u

∂α

∂t

⟩dt

∣∣∣∣u=0

=∫S1

[⟨D

∂u

∂α

∂t,D

∂u

∂α

∂t

⟩+⟨∂α

∂t,D2

∂u2

∂α

∂t

⟩]dt

∣∣∣∣u=0

,

where D denotes the covariant derivative of the Levi-Civita connection on theambient Riemannian manifold M . Thus

d2(J)(γ)(V, V ) =∫S1

[⟨DV

∂t,DV

∂t

⟩+⟨∂γ

∂t,D

∂u

DV

∂t

⟩]dt

Using the definition of the Riemann-Christoffel curvature tensor R, we obtain

d2Jγ(V, V ) =∫S1

[⟨DV

∂t,DV

∂t

⟩+ 〈γ′(t), R(V, γ′(t))V 〉+

⟨γ′(t),

D

∂t

DV

∂u

⟩]dt.

An integration by parts and use of the fact that γ satisfies the Euler-Lagrangeequation eliminates the last term, thereby yielding (2.7).

We can write the formula for second variation as

d2J(γ)(V,W ) =∫

Σ

〈L(V ),W 〉dt,

where Lγ is the Jacobi operator, defined by

L(V ) = −[D

∂t DV∂t

+R(V, γ′(t))γ′(t)]. (2.8)

An element V ∈ TγM is called a Jacobi field along γ if Lγ(V ) = 0. Note thatif γ ∈ C∞(S1,M), then the Jacobi operator yields a continuous linear operator

L : L2k(γ∗TM))→ L2

k−2(γ∗TM)),

for all k ≥ 1. Symmetry of d2J implies that the Jacobi operator satisfies∫S1〈L(V ),W 〉dt =

∫S1〈V,L(W )〉dt, for all V,W ∈ L2

k+2(γ∗TM)),

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and hence we say that it is formally self-adjoint . In studying the Hessian of Jit is also useful to consider, for λ ∈ C, the closely related operator

Lλ = L− λι : L2k(γ∗TM))→ L2

k−2(γ∗TM)), (2.9)

where ι : L2k(γ∗TM)) → L2

k−2(γ∗TM)) is the inclusion. We say that λ is aneigenvalue for L if the kernel of Lλ is nonzero.

There is a similar second variation formulae for J : M → R when M =Ω(M ; p, q). In this case, we obtain

d2J(γ)(V, V ) =∫ 1

0

[⟨DV

∂t,DV

∂t

⟩+ 〈γ′(t), R(V, γ′(t))V 〉

]dt,

except that now V is an element of

TγΩ(M ; p, q) = V ∈ L21([0, 1], TM) : ωπ(V ) = γ, V (0) = 0 = V (1).

In this case an integration by parts shows that

d2J(γ)(V,W ) =∫ 1

0

〈L(V ),W 〉dt, for V,W ∈ TγΩ(M ; p, q), (2.10)

where L is the differential operator defined once again by (2.8). If we set

L21,0(γ∗TM) = X ∈ L2

k(γ∗TM)) : X(0) = X(1) = 0,

then the Jacobi operator defines a continuous linear map

L : L21,0(γ∗TM)) ∩ L2

k(γ∗TM))→ L2k−2(γ∗TM)),

for all k ≥ 1. Once again, we can define, for λ ∈ C, the closely related operator

L : L21,0(γ∗TM)) ∩ L2

k(γ∗TM))→ L2k−2(γ∗TM)), (2.11)

where ι is the inclusion.The Jacobi operators L are special cases of formally self-adjoint elliptic differ-

ential operators and there is a well-developed classical theory for the eigenvalueproblem for such operators. In order to describe this theory, we need the notionof Fredholm operator:

Definition. A linear operator L : E → F between Banach spaces is said to beFredholm if

1. L has finite-dimensional kernel,

2. L has closed range, and

3. L has finite-dimensional cokernel, where the cokernel of L is the quotientspace F/(Range(L)).

77

The Fredholm index of a Fredholm operator L is defined by the formula

Fredholm index of L = dim(Kernel of L)− dim(Cokernel of L).

Note that a linear operator between finite-dimensional Banach spaces, say L :Rn → Rp, is always Fredholm, and its Fredholm index is just the difference indimensions n− p.

Theorem 2.4.2 For every λ ∈ C and every integer k ≥ 1, the operators Lλ de-fined by (2.9) and (2.11) are Fredholm maps of Fredholm index zero. Moreover,

1. for each λ ∈ C the eigenspace Wλ = Ker(Lλ) is finite-dimensional.

2. all the elements of Wλ are C∞,

3. if Wλ is empty, then Lλ possesses an inverse Gλ, which is called a Green’soperator,

4. if Wλ is nonempty, that is, λ is an eigenvalue, then λ ∈ R, and

5. the eigenvalues can be arranged in a sequence

λ1 < λ2 < · · · < λi < · · · with λi →∞,

and only finitely many of the eigenvalues are negative.

This is a special case of the basic theorem for second-order elliptic operatorsproven in basic courses on linear PDE’s. It is worked out in various cases inChapter 5 of [79], Chapter 3 of [44] and in [19].

2.5 Morse nondegenerate critical points

IfM is a Hilbert manifold with a Riemannian metric 〈〈·, ·〉〉 and f :M→ R is aC2 map with a critical point p, it follows from the Riesz representation theoremthat there is a continuous map A : TpM→ TpM such that

d2f(p)(V,W ) = 〈〈A(V ),W 〉〉, for V,W ∈ TpM, (2.12)

Definition. Suppose that M is a Hilbert manifold and that f : M → R is aC2 map. A critical point p for f is Morse nondegenerate if for some choice (andhence every choice) of Riemannian metric 〈〈·, ·〉〉 the continuous map A : TpM→TpM satisfying (2.12) is an isomorphism; otherwise, it is Morse degenerate.

Example 2.5.1. A Morse nondegenerate critical point has zero nullity, but theconverse is not true in general. Indeed, we can let M = L2[0, 1] with the innerproduct

〈φ, ψ〉 =∫ 1

0

φ(t)ψ(t)dt,

78

and define

f : L2[0, 1]→ R by f(φ) =∫ 1

0

tφ(t)2dt.

Then φ = 0 is a critical point for f and

d2f(0)(φ, ψ) = 〈A(φ), ψ〉, where A(φ(t)) = tφ(t).

Then A is injective but the range of A does not include any constant functions,so it is not surjective. Thus the critical point 0 has zero nullity but is not Morsenondegenerate.

In the variational problems we have been considering, however, the map A isFredholm and zero nullity does indeed imply nondegeneracy.

To be specific, let’s focus on the case where M is a complete Riemannianmanifold, properly and isometrically imbedded in RN andM = Ω(M,p, q). ForRiemannian metric on M, we choose the intrinsic Riemannian metric 〈〈·, ·〉〉defined by the formula

〈〈V,W 〉〉γ =∫ 1

0

[〈V (t),W (t)〉+

⟨DV

∂t(t),

DW

∂t(t)⟩]

dt, (2.13)

for V,W ∈ TγΩ(M,p, q), where D is the Levi-Civita connection on M . Anintegration by parts shows that

〈〈V,W 〉〉γ =∫ 1

0

〈P (V ),W 〉dt, where P = −Ddt Ddt

+ id,

a formally self-adjoint second order elliptic partial differential operator. Sincethe Hilbert space inner product is positive definite, it follows from Theorem 2.4.2that

P : L21,0(γ∗TM)→ L2

−1(γ∗TM)

has an inverse Green’s operator G.Now consider the action function J : Ω(M,p, q) → R. It follows from the

second variation formula (2.10) that

d2J(γ)(V,W ) =∫ 1

0

〈L(V ),W 〉dt = 〈〈G L(V ),W 〉〉 = 〈〈A(V ),W 〉〉,

where A is a Fredholm operator of Fredholm index zero. Thus if γ is a criticalpoint for J with zero nullity, then γ is a Morse nondegenerate critical point.

Definition. We say that f : M → R is a Morse function if all of its criticalpoints are Morse nondegenerate.

Theorem 2.5.2. Suppose that (M, 〈·, ·〉) is a complete Riemannian manifoldand p ∈M . For almost all choices of q ∈M , the function

J : Ω(M ; p, q) −→ [0,∞)

79

is a Morse function.

The proof follows from the fact that there are nonzero Jacobi fields along γ ∈Ω(M ; p, q) vanishing at the endpoints if and only if q is a critical value for expp.Moreover, it follows from Sard’s Theorem (which is proven in [36], Chapter 3)that the set of regular values for expp form a set of full measure in M . Thus foralmost all choices of q, all geodesics in Ω(M ; p, q) will be Morse nondegenerate.

In the case where M is a compact Riemannian manifold and M = L21(S1,M),

we utilize the similar Riemannian metric 〈〈·, ·〉〉 defined by

〈〈V,W 〉〉γ =∫S1

[〈V (t),W (t)〉+

⟨DV

∂t(t),

DW

∂t(t)⟩]

dt,

for V,W ∈ TγL21(S1,M)). Once again, we can integrate by parts to obtain

〈〈V,W 〉〉γ =∫S1〈P (V ),W 〉dt, where P = −D

dt Ddt

+ id,

and P has a Green’s operator inverse G. The operator A = G L satisfying

d2J(p)(V,W ) = 〈〈A(V ),W 〉〉, for V,W ∈ TpL21(S1,M)

is once again a Fredholm operator of Fredholm index zero.For the space of free loops, however, J : L2

1(S1,M) → R is never a Morsefunction. The reason is that the action J is invariant under a continuous rightaction of the circle group S1 on L2

1(S1,M),

φ : L21(S1,M)× S1 −→ L2

1(S1,M), φ(γ, s) = γs, where γs(t) = γ(t+ s),

and therefore whenever any point is critical for J , so is the entire S1-orbit. Theaction J is also invariant under an extension of this action to an O(2)-action

φ : L21(S1,M)×O(2) −→ L2

1(S1,M),

the reflections in O(2) changing the orientation of the geodesic critical points.However, after we describe Smale’s extension of Sard’s theorem to infinite-dimensional manifolds in the next section, the theory of Fredholm maps willallow us to perturb J to a Morse function.

We have seen that we needed to choose the L21 topology on the space of

maps in order for Condition C to hold, but we also know that for the varia-tional problems coming from simple mechanical systems, the critical points areautomatically C∞. For many purposes it is convenient to have extra derivatives,and hence to work in the space of L2

k maps, for k > 1. This is admissible solong as we don’t utilize convergence results that rely upon condition C.

For example, we can take M = L2k(S1,M) with underlying Riemannian

manifold (M, 〈·, ·〉), and consider a function f = J :M→ R of the form

J(γ) =∫

Σ

L(γ)dt, where L : L2k(S1,M)→ L1(S1,M)

80

is a sufficiently well-behaved function, called the Lagrangian of the variationalproblem. We assume that we can differentiate to obtain

(dJ)(γ)(V ) =∫S1〈F (γ), V 〉dA, (2.14)

for some function F . The equation F (γ) = 0 is called the Euler-Lagrange equa-tion for the variational problem, and F is called the Euler-Lagrange operator .For example, in the case of a simple mechanical system (M, 〈·, ·〉, φ), we take

L(γ) =12〈γ′(t), γ′(t)〉 − φ(γ(t)),

and the corresponding Euler-Lagrange operator is just

F (γ) = −∇′γγ′ − (grad φ)(γ′(t)),

a nonlinear second-order differential operator.To have an appropriate range for the nonlinear differential operator F it

is convenient to utilize pullback bundles. For k ∈ N, k > 2, we can regardL2k−2(S1, TM) as the total space of a smooth vector bundle over L2

k−2(S1,M),and let L2

k−2(S1, TM) denote the total space of the pullback bundle via theinclusion ι to L2

k(S1,M), so

L2k−2(S1, TM) = (γ, V ) ∈ L2

k(S1,M)× L2k−2(S1, TM) : X ∈ L2

k−2(γ∗TM).(2.15)

However, note that the explicit construction (2.15) makes sense even if k = 2or k = 1. Thus the Euler-Lagrange map F can be regarded as a map

F : L2k(S1,M)→ L2

k−2(S1, TM),

for all k ∈ N, which is differentiable when k is large. If γ is a critical point forJ , we can differentiate to obtain a formula for the Hessian,

d2J(γ)(V,W ) =∫

Σ

〈(πV DF )(γ)(V ),W 〉dA, (2.16)

where DF denotes the derivative with respect to γ ∈ L2k(Σ,M) and πV is the

vertical projection into the fiber L2k−2(γ∗TM). Of course, L = πV (DF )(γ) is

just the Jacobi operator at γ. The formula shows that we can regard the Jacobioperator L as the linearization of the Euler-Lagrange operator F at a solutionγ to the Euler-Lagrange equation.

Recall that the vector field X = −grad(J) on L21(S1,M) is defined by the

equation〈〈X (γ), V 〉〉 = −dJ(γ)(V ), for V ∈ TγL2

1(S1,M). (2.17)

Thus the linearization of −X at a critical point γ ∈ L2k(S1,M) (when k > 2) is

a smoothed version of the Jacobi operator. Indeed, since F (γ) = 0, it followsfrom (2.14) that

−πV DX (γ) = G πV DF (γ) = G L = A,

81

where A is a restriction of the Fredholm operator satisfying (2.12).

Remark 2.5.3. The vector field X = −grad(J) is tangent to each of thesubpaces L2

k(S1,M) ⊆ L21(S1,M). Indeed, as mentioned before,

〈〈V,W 〉〉γ =∫S1〈Pγ(V ),W 〉dt, where Pγ = −D

dt Ddt

+ id.

As γ varies, the differential operators Pγ fit together to form a vector bundlemap

P : L2k(S1, TM)→ L2

k−2(S1, TM),

where L2k−2(S1, TM) is the total space of a vector bundle over L2

k(S1,M) asdescribed in (2.15). The vector bundle map P has a “Green’s operator” inverse

G : L2k−2(S1, TM)→ L2

k(S1, TM),

a smoothing operator which increases the number of derivatives by two. If

F : L2k(S1,M)→ L2

k−2(S1,M)

is the Euler-Lagrange map for Jψ, then (2.17) implies that

X = −G F : L2k(S1,M)→ L2

k−2(S1, TM).

2.6 The Sard-Smale Theorem

Sard’s theorem, so useful in understanding the transversality theory of finite-dimensional manifolds, possesses an extension to infinite-dimensional manifoldsthat is due to Smale. The standard approach to constructing Morse functionson infinite-dimensional manifolds is based upon the Sard-Smale theorem.

Recall that a point q ∈ M2 is called a regular value for the smooth mapf :M1 →M2 if

p ∈ f−1(q) ⇒ dfp is onto;

otherwise it is called a critical value. Any finite-dimensional manifold has ameasure which defines volume integrals on open subsets of M ; indeed, if D isan open subset of a coordinate chart (U, (x1, . . . xn)) on M , we can set

Volume of D =∫D

√gdx1 · · · dxn.

Here g is the determinant of the component matrix of the metric tensor, and itis standard to check that the integral for volume is independent of coordinatechart chosen. Using the Riesz representation theorem from analysis, one canthen define measurable subsets of M and speak of sets of measure zero.

Brown-Sard Theorem 2.6.1. Suppose that f : M1 → M2 is a Ck mapbetween finite-dimensional manifolds, where

k > 0 and k > dim(M1)− dim(M2).

82

Then the subset of M2 consisting of the critical values of f has measure zero.

A proof of this theorem can be found in [36], Chapter 3. The strange differen-tiability condition cannot be weakened to k ≥ dim(M1)− dim(M2). However,this finite-dimensional theorem actually does have a slight improvement thatwas found by Bates [6]. Let Ck,1 denote the class of functions which are Ck andmoreover have k-th order derivatives which are Lipshitz. Bates showed that iff : M1 →M2 is Ck,1, where

k > 0 and k ≥ dim(M1)− dim(M2),

then the set of critical values of f has measure zero.

Suppose now that M1 and M2 are manifolds modeled on Banach spaces E1

and E2 respectively. A Ck map f : M1 → M2, where k ≥ 1, is said to be aFredholm map if its linearization

(f∗)p : TpM1 −→ Tf(p)M2

is a Fredholm operator, for each p ∈ M1. The Fredholm index of a Fredholmmap f :M1 →M2 is the Fredholm index of (f∗)p for any p ∈ M1, this beingconstant ifM1 is connected. Indeed, it can be proven that the Fredholm indexis a continuous function from the space of Fredholm operators to the integers.

Note that ifM1 andM2 are finite-dimensional, any C1 map f :M1 →M2

is Fredholm with Fredholm index dim(M1)− dim(M2).Thus the Fredholm index can be used to replace one of the hypotheses in the

Brown-Sard Theorem. We also need to replace the notion of “measure zero,”since we have no completely satisfactory notion of measure in infinite dimensionspossessing all of the nice properties of the standard measure on submanifoldsof finite-dimensional Euclidean space. We say that a subset of M2 is residualor generic if it is a countable intersection of open dense subsets of M2. Withthese preparations out of the way, we can now state Smale’s version of Sard’stheorem [78]:

Sard-Smale Theorem 2.6.2. Suppose that f :M1 →M2 is a Ck Fredholmmap between separable Banach manifolds, where

k > 0 and k > the Fredholm index of f .

Then the set of regular values of f is residual.

The main idea of Smale’s proof is to reduce to the finite-dimensional case. Wefirst need to show that a Fredholm map is locally proper:

Lemma 2.6.3. If f :M1 →M2 is a Fredholm map and p ∈M1, then there isan open neighborhood U of p such that the restriction of f to the closure U ofU is proper and closed.

To prove the Lemma, we use the “local representative theorem” for f near p,a direct corollary of the inverse function theorem. According to this theorem,

83

there exist direct sum decompositions of the model spaces for M1 and M2:

E1 = E ⊕G1, E2 = E ⊕G2,

where G1 and G2 are both finite-dimensional, and local charts (U, φ) on M1

and (V, ψ) on M2 centered at p and f(p) respectively, such that

ψ f φ−1 : φ(U ∩ F−1(V )) −→ E2

is of the special form

ψ f φ−1(u, v) = (u, η(u, v)), for u ∈ E, v ∈ G1,

where Dη(0, 0) = 0. We denote ψ f φ−1 simply by f .Suppose now that D1 and D2 are closed balls of radius ε > 0 in E and G1

respectively. To show that f |(D1×D2) is proper, we suppose that pi = (ui, vi) ∈D1 ×D2 and f(pi) = qi → q ∈ V . We need to show that pi has a convergentsubsequence. Since D2 is compact, we can assume that vi → v ∈ D2. On theother hand,

f(ui, vi) = (ui, η(ui, vi))→ q ⇒ ui → some u ∈ D1.

Hence pi → (u, v) ∈ D1 ×D2. It follows that f |(D1 ×D2) is proper.To prove that f |(D1×D2) is closed, we suppose that pi = (ui, vi) ∈ D1×D2

converges to p = (u, v). Then ui converges to some point u and η(ui, vi) hasa subsequence which converges to some point w. Then f(ui, vi) converges to(u,w).

Returning to the proof of the theorem, we note that since M1 is separable, theLemma implies that it can be covered by a countable collection of open sets Uisuch that f |U i is proper. Thus we can reduce the proof to the case where fis proper and closed. But this implies that the set of critical values is closed.Thus in this case the regular values form an open set, and it suffices to provethat the regular values are dense. In other words, it suffices to show that anyopen subset of M2 contains a regular value.

It suffices to show that there is a regular value in a product neighborhood Uiconstructed as in the proof of the lemma. It follows from the finite-dimensionalversion of Sard’s theorem that for each fixed choice of u0, there is a regularvalue v0 for

v 7→ ηi(u0, v),

and (u0, v0) is then a regular value, proving the theorem.

Note that by Corollary 1.2.8, if f :M1 →M2 is a Ck Fredholm map betweenseparable Banach manifolds of Fredholm index m, where m < k, then wheneverq is a regular value of f , f−1(q) is a submanifold of of M1 of dimension m.

84

2.7 Existence of Morse functions

We now consider an important application of the Sard-Smale theorem, the per-turbation of a given function to a Morse function.

In the case of the action integral, the circle group S1 acts on the free loopspace L2

1(S1,M) preserving the action J , and this ensures that all critical pointsfor J must be Morse degenerate, and in fact that all nonconstant multiples of γ′

are Jacobi fields along γ. However, there is a simple perturbation of the actionintegral whose critical points are all Morse nondegenerate, and this perturbationis therefore a Morse function. The perturbation is obtained by making a suitablechoice of ψ ∈ Ck(S1,RN ) for k sufficiently large, and setting

Jψ : L21(S1,M)→ R by Jψ(γ) =

12

∫S1‖γ′(t)‖2dt+

∫S1γ(t) · ψ(t),

where the dot denotes the usual dot product in the ambient Euclidean space.In this section, we will show that for most choices of ψ, Jψ is indeed a Morsefunction. Recall that a subset of a complete metric space is residual if it is thecountable intersection of open dense sets.

Theorem 2.7.1. For a residual set of ψ ∈ L21(S1,RN ), the function

Jψ : L21(S1,M)→ R

is a Morse function; that is, all of its critical points are nondegenerate.

Note. Since critical points of Jψ are automatically elements of L23(S1,M)

when ψ ∈ L21(S1,RN ), it suffices to show that Jψ : L2

3(S1,M) → R is a Morsefunction.

Proof: It is readily verified that γ is a critical point for Jψ if and only if

dJψ(X) = 0, for all X ∈ Tγ(L23(S1,M)).

We can writedJψ(γ)(X) =

∫Σ

〈F (γ, ψ), X〉dA,

where F (γ, ψ) = 0 is the Euler-Lagrange equation for Jψ, so that γ is a criticalpoint for Jψ if and only if F (γ, ψ) = 0.

A direct calculation shows that the Euler-Lagrange map

F : L23(S1,M)× L2

1(S1,RN )→ L21(S1, TM) is given by

F (γ, ψ) = −∇γ′γ′ + ψT ,

where ψT denotes the orthogonal projection of ψ into the tangent space ofM . We can regard F as a vector field on L2

3(S1,M), depending on a param-eter in L2

1(S1,RN ), which loses two derivatives, and hence takes its values inL2

1(S1, TM):

−∇γ′γ′ + ψT ∈ X ∈ L21(S1, TM) : ωπ X = γ.

85

Lemma 2.7.2. F is transverse to the zero-section of the vector bundle

ωπ : L21(S1, TM) −→ L2

1(S1,M).

Proof of Lemma: Taking the partial derivative with respect to the second vari-able, we obtain πV (D2F )(γ, ψ)(η) = −ηT , where πV denotes projection intothe vertical tangent space at (γ, 0) ∈ L2

1(S1, TM). The formula shows thatπV (D2F )(ψ, γ) maps onto the fiber of ωπ over γ. Hence if γ is a critical pointfor Jψ,

(Range of πV (D2F )(ψ, γ)) + (Tangent space to zero-section)= Tangent space to L2

1(S1, TM),

which means that F is transverse to the zero-section.

Just as in the finite-dimensional case, it follows from Corollary 1.2.8 that theinverse image of a split submersion of a submanifold is itself a submanifold, soF−1(zero-section) is a submanifold S of L2

3(S1,M)× L21(S1,RN ). We can also

describe this submanifold as the solution set

S = (γ, ψ) ∈ L23(S1,M)× L2

1(S1,RN ) : ∇γ′γ′ − ψT = 0.

Lemma 2.7.3. The projection on the second factor π : S → L21(S1,RN ) is a

Fredholm map of Fredholm index zero.

We begin by determining the tangent space to S, obtaining

T(γ,ψ)S = (X, η) ∈ TL23(S1,M)× L2

1(S1,RN ) :πV D1F (γ, ψ)X + πV D2F (γ, ψ)η = 0,

or equivalently,

T(γ,ψ)S = (X, η) ∈ TL23(S1,M)× L2

1(S1,RN ) : Lγ,ψ(X) = −ηT , (2.18)

whereLγ,ψ = πV D1F (γ, ψ) : TγL2

3(S1,M)→ TγL21(S1,M)

is the Jacobi operator for Jψ determined by the formula

d2Jψ(γ)(X,Y ) =∫

Σ

〈Lγ,ψ(X), Y 〉dA, for X,Y ∈ TγL23(S1,M).

Let B : TM × TM → NM denote the second fundamental form of M in RNand define for n ∈ NpM ,

An : TM → TM by 〈An(x), y〉 = B(x, y) · n.

86

Then a calculation (similar to that used to prove Proposition 2.4.1) shows thatthe Jacobi operator is given by the formula

Lγ,ψ(X) = −∇γ′∇γ′(X)−R(X, γ′)γ′ −Aψ(t)⊥(X). (2.19)

One readily verifies that Lγ,ψ is a Fredholm operator of Fredholm index zero.We next calculate the kernel and image of the map dπ(γ,ψ) : Tγ,ψS →

L21(S1,RN ) and after a short derivation, we obtain the results:

Kernel of dπ(γ,ψ) = (X, η) ∈ T(γ,ψ)S : η = 0, Lγ(X) = 0,

Range of dπ(γ,ψ) = Lγ,ψ(X) + χ⊥ : X ∈ TγL23(S1,M), χ ∈ L2

1(S1,RN ),

where χ⊥ denotes the orthogonal projection of ψ into the normal space to M .Note that the kernel of dπ(γ,ψ) is isomorphic to the kernel of Lγ,ψ while therange of dπ(γ,ψ) is a subspace of L2

1(S1,RN ) which has the same codimensionas Range(Lγ,ψ) ⊂ TγL2

1(S1,M).Thus dπ(γ,ψ) is a Fredholm operator with the same Fredholm index as Lγ,ψ,

namely zero, finishing the proof of Lemma 2.6.3.

According to the Sard-Smale Theorem 2.6.2, there is a countable intersectionof open dense subsets of ψ ∈ L2

2(S1,RN ) consisting of regular values of π. Butif ψ is a regular value for π, then Kernel(Lγ,ψ) = 0 at each critical point, so allcritical points are Morse nondegenerate, and Jψ is a Morse function, establishingTheorem 2.7.1.

Definition. Suppose that M is a Banach manifold and f : M → R is a C2

function. A compact finite-dimensional submanifold N of M is said to be anondegenerate critical submanifold for f if

1. every p ∈ N is a critical point for f .

2. if p ∈ N , then TpN is the set of V ∈ TpM such that d2f(p)(V,W ) = 0,for all W ∈ TpM.

The Morse index of a connected nondegenerate critical submanifold N is theindex of any critical point p ∈ N .

Proposition 2.7.4. If M is a compact Riemannian manifold and

M0 = γ ∈ L21(S1,M) : γ is constant ,

then M0 is a nondegenerate critical submanifold of L21(S1,M) of Morse index

zero. Moreover, for some ε > 0, the open neighborhood

Uε = γ ∈ L21(S1,M) : |γ −M0|C0 < ε

has M0 as a strong deformation retract and contains no critical points for theaction J other than the elements of M0.

87

To prove this, first note that if γp denotes the constant loop at the point p ∈M ,then

TγpL21(S1,M) ∼= L2

1(S1, TpM)

and the tangent subspace TγpM0 consists of the constant maps into TpM . Onthe other hand, the normal space to M0 at γp can be identified with

NγpM0 =V ∈ L2

1(S1, TpM) :∫S1V (t)dt = 0

.

According to the second variation formula (2.7),

d2J(γp)(V,W ) =∫S1〈V ′(t),W ′(t)〉dt, for V,W ∈ L2

1(S1, TpM),

from which we conclude that d2J(γp)(V,W ) = 0 if V is constant, that is, if Vlies in TγpM0. On the other hand, it is readily verified that d2J(γp) is positivedefinite on NγpM0, so M0 is indeed a nondegenerate critical submanifold ofMorse index zero.

Let NM0 denote the total space of the normal bundle of M0 and for ε > 0,let

Vε = V ∈ NγpM0 for some p ∈M such that |V |C0 < ε ,and define

φ : Vε → Uε by φ(V ) = expp(V ).

For ε > 0 sufficiently small, we can define

H : Uε × [0, 1]→ Uε by H(expp(V ), t) = expp(tV ).

Then H is smooth, H(γ, 1) = γ for γ ∈ Uε, H(γ, t) = γ for γ ∈ M0 and allt ∈ [0, 1], so γ 7→ H(γ, 0) is the desired deformation retraction from Uε to M0 .Finally, using a Taylor expansion about points in M0, one verifies that Uε−M0

contains no critical points for J .

We can now alter the definition of Jψ slightly. Suppose that η : R → [0, 1] is asmooth function such that

η(t) =

0, for t ≤ ε2/4,1, for t ≥ ε2/2.

Note that since L(γ)2 ≤ 2J(γ), η(J(γ)) = 1 outside Uε. Given a smooth mapψ : S1 → RN , we consider the function Jψ defined by

Jψ(γ) = J(γ) + η(J(γ))∫S1

(γ · ψ)dt. (2.20)

It follows from Theorem 2.7.1 that if ψ is generic and sufficiently small, thenJψ ≥ 0 and M0 = J−1

ψ (0) is a critical submanifold of Morse index zero. More-over, all other critical points of Jψ are Morse nondegenerate, so the restrictionof Jψ to L2

1(S1,M)−M0 is a Morse function.

88

2.8 Bumpy metrics for smooth closed geodesics*

With additional work, one can show that if M is a compact manifold, thenfor generic choice of Riemannian metric on M , all nonconstant smooth closedgeodesics have the property that their only Jacobi fields are those generated bythe S1 action on L2

1(S1,M). This is the “Bumpy Metric Theorem” of Abraham,and a proof due to Anosov is found in [4]; an alternate proof can be found in[7]. We will present another proof in this section, based upon Bott’s theory ofiterated smooth closed geodesics [8]. The proof is long so some readers maywant to skip it on first reading.

We first prove a simpler version for the case where the geodesic is prime,that is, not a nontrivial cover of a geodesic of smaller length:

Theorem 2.8.1. If M is a compact manifold, then for generic choice of Rie-mannian metric on M , all nonconstant prime smooth closed geodesics have theproperty that their only Jacobi fields are those generated by the S1 action onL2

1(S1,M).

Proof of Theorem 2.8.1: The techniques are exactly the same as those usedbefore. Note first that since critical points for the action function J are smooth,we are not restricted to working in L2

1(S1,M), but can work for example in aSobolev space of more highly differentiable functions, such as L2

2(S1,M), whichis also a Hilbert manifold.

We need a preliminary step in the argument to deal with the fact that J isG-equivariant, where G = S1. If N is a compact codimension one submanifoldof M with boundary ∂N , we let

U(N) = nonconstant γ ∈ L2k(S1,M) : γ does not intersect ∂N

and has transversal intersection with the interior of N,

an open subset of L22(S1,M). We cover L2

2(S1,M) with a countable collectionUi = U(Ni) of open sets corresponding to a countable collection of codimensiontwo submanifolds Ni. If 0 is a choice of origin in S1, we let

Fi = γ ∈ U(Ni) : γ(0) ∈ Ni,

a C1 submanifold of L22(S1,M) by the smoothness theorems in §1.6. Note that

Fi meets each nonconstant G-orbit in U(Ni) in a finite number of points.LetM denote the manifold of L2

` Riemannian metrics on M , an open subsetof the space of L2

` sections of the second symmetric power of T ∗M , where ` isa large integer. We claim that

S = (γ, g) ∈ Fi ×M : γ is a geodesic for the metric g

is a smooth submanifold of Fi×M. To see this, we let L2∗(S

1, TM) denote thevector bundle over Fi whose fiber at each f ∈ L2

2(S1,M) is L2(S1, γ∗TM). Itfollows from multiplication theorems from the theory of Sobolev spaces (L2

2·L2 ⊂

89

L2) that L2∗(S

1, TM) does in fact have the structure of a smooth vector bundleover Fi. We next define a C1 map

F : Fi ×M −→ L2∗(S

1, TM) by F (γ, g) = ∇gγ′γ′,

where ∇g is the Levi-Civita connection on TM determined by the metric g onM . Let Z denote the image of the zero section of L2

∗(Σ, TM) and note thatS = F−1(Z). Our goal is to show that F is as transverse to Z as possible.

The first derivative of the action is given by the formula

dJ(γ)(X) =∫S1〈F (γ, g), X〉dt.

Differentiating once again gives us the Hessian at a critical point,

d2J(γ)(X,Y ) =∫S1〈D1F (γ, g)(X), Y 〉dt =

∫S1〈Lγ(X), Y 〉dt, (2.21)

where D1F denotes derivative with respect to the first variable

γ ∈ L22(S1,M) : γ(0) ∈ Ni,

and

Lγ : X ∈ L22(S1, γ∗TM) : X(0) ∈ Tγ(0)Ni −→ L2(S1, γ∗TM) (2.22)

is the Jacobi operator. Note that at a zero (γ, g) of F the tangent space toL2∗(S

1, TM) can be divided into a direct sum

Tγ(L2∗(S

1, TM)) = H ⊕ V,

where H is horizontal (tangent to the zero section) and V is vertical (tangentto the fiber). If πV denote the projection onto V along H, it follows from (2.21)that

πV (D1F )(γ,g)(X) = Lγ(X). (2.23)

Because it is a Jacobi field, the section T (t) = γ′(t) is not in the image of Lγ .However, we claim that

the image of πV DFγ,g ⊕ (span of T (t)) = L2(S1, γ∗TM), (2.24)

or equivalently, that V is spanned by T (t), where

V = L2-sections X of γ∗TM : X is ⊥ to the image of πV DFγ,g.

To show this, we need to calculate πV (D2F ), where D2F is the partial deriva-tive with respect to the second variable g ∈M.

Suppose that γ is a nonconstant geodesic, which will be C` when the metricis L2

` . Choose a point t ∈ S1 which possesses a neighborhood U such that γimbeds U into some open set W ⊂M on which Fermi coordinates (x1, . . . , xn)are defined. Such coordinates satisfy the following conditions:

90

1. γ is described by the equations x2 = · · · = xn = 0,

2. x1 γ = t,

3. the metric g takes the form∑gijdu

iduj , such that on γ(U), gij = δij , for1 ≤ i, j ≤ n.

In terms of these local coordinates, the equation for geodesics becomes

d2xkdt2

+∑i,j

Γkijdxidt

dxjdt

= 0.

The expression on the left side of this equation is called the acceleration of thepath.

A perturbation in the metric h ∈ TgM with compact support in W can bewritten in the form h =

∑hijdxidxj . Under this perturbation, the only piece

of the acceleration that changes is the Christoffel symbol

Γkij =12

∑gkl(∂gil∂xj

+∂gjl∂xi− ∂gij∂xl

),

and if Γkij denotes the derivative of Γkij in the direction of the perturbation,

Γkij =12

(∂hik∂xj

+∂hjk∂xi

− ∂hij∂xk

).

We then find that

πV (D2F )(γ,g)(h) =n∑

i,j,k=1

Γkijdxidt

dxjdt

∂

∂xk=

n∑k=1

Γk11

∂

∂xk. (2.25)

We can select the perturbation so that

h11 = x2φ, hij = 0, for other choices of indices i, j,

where φ has compact support in W and 2 ≤ r ≤ n; we then see that

Γr11(t, 0, . . . , 0) = −12∂

∂x2(h11)(t, 0, . . . , 0) = −1

2φ(t, 0, . . . , 0).

Thus the fiber projection of the partial derivative of F with respect to the secondvariable (in M) is given by the expression

πV (D2F )f,g(h) = −12

n∑r=2

Γr11

∂

∂xr= −1

2φ∂

∂x2.

In this manner, we can show that any vector field of the form

n∑r=2

φr∂

∂xr

91

where φr is a smooth function with compact support, lies in the image of πV D2Ff,g, and hence elements of V must be tangent to γ in U .

Since a dense open subset of points of Σ can be covered by sets of the formU , we see that elements of V must be tangent to γ over all of S1. On the otherhand, they must also be image of Lγ restricted to complement of the span ofT (t) by (2.23). Thus we obtain the desired result (2.24).

The remainder of the proof is similar to the proof of Theorem 2.7.1. Notethat the tangent space to S is given by the formula

Tγ,gS = (X,h) ∈ TγFi × TgM : Lγ(X) + πV (D2F )(h) = 0,

which makes it easy to analyze the kernel and range of the map dπ(γ,g) :T(γ,g)S → TgM. Indeed,

Kernel of dπ(γ,g) = (X,h) ∈ TγFi × TgM : h = 0, Lγ(X) = 0,

while

Range of dπ(γ,g) = h ∈ TgM; there exists an element X ∈ TγL22(S1,M)

such that X(0) ∈ Tγ(0)Ni and Lγ(X) = −πV (D2F )(h).

Thus the kernel of dπ(γ,g) is isomorphic to the kernel of Lγ . On the other hand,if h is an element of TgM such that

πV (D2F )(h) = 0, then (0, h) ∈ T(γ,g)S,

and hence h lies in the range of dπ(γ,g). It follows that complement to the rangeof dπ(γ,g) must inject to a complement to the range of Lγ , and in particular,any such complement is finite-dimensional, so dπ(γ,g) is a Fredholm map, andthe dimension of the cokernel of dπ(γ,g) is no larger than the dimension of thecokernel of Lγ . By the earlier transversality argument, dFγ,g maps surjectivelyonto a complement to the one-dimensional space generated by the Jacobi fieldT (t), so the dimension of the cokernel of dπ(γ,g) is actually one less than thedimension of the cokernel of Lγ .

Thus the Fredholm index of dπ(γ,ψ) one more than the Fredholm index of themap Lγ of (2.22), which is one because the restriction X(0) ∈ Tγ(0)Ni cuts downthe kernel by one. It follows that the Fredholm index of dπ itself is zero. Thuswe can use the Sard-Smale Theorem to show that a countable intersection ofopen dense subsets ofM consist of regular values for π. If g0 is such a “genericmetric,” all of the prime geodesics will be Morse nondegenerate, finishing theproof of the Bumpy Metric Theorem for prime geodesics.

Implication of the above Theorem: It follows from Theorem 2.8.1, togetherwith condition C of Palais and Smale, that for a Riemannian metric belongingto a residual subset, the number of S1-orbits of nonconstant prime geodesics oflength less than a given bound is finite.

We will next extend the argument from the previous theorem to cover the casein which geodesics are not necessarily prime, thereby obtaining:

92

Bumpy Metric Theorem 2.8.2. If M is a compact manifold, then for genericchoice of Riemannian metric onM , all nonconstant smooth closed geodesics havethe property that their only Jacobi fields are those generated by the S1 actionon L2

1(S1,M).

Of course a geodesic which is not prime covers a prime minimal geodesic andour strategy is to study this underlying prime geodesic.

We make the following definitions following Bott [8]. Suppose that γ : R→M is a smooth geodesic with γ(t+ 2π) = γ(t). If q ∈ Z and z ∈ S1 ⊂ C, we let

V∞q,z = smooth sections X of γ∗TM ⊗ C : X(t+ 2πq) = zX(t),

and define an inner product

〈·, ·〉q : V∞q,z × V∞q,z −→ C

by

〈X, Y 〉q =∫ 2πq

0

[⟨DX

∂t,DY

∂t

⟩+ 〈X, Y 〉

]dt,

where D denotes the covariant derivative defined by the Levi-Civita connectionon M . Let Vq,z denote the completion of V∞q,z with respect to 〈·, ·〉q and define

Iq(·, ·) : Vq,z × Vq,z −→ C

by

Iq(X, Y ) =∫ 2πq

0

[⟨DX

∂t,DY

∂t

⟩− 〈R(X, γ′)γ′, Y 〉

]dt,

which is of course the restriction of d2J(γ) to Vq,z.

Lemma 2.8.3. The inclusion ∑zq=1

V1,z ⊂ Vq,1 (2.26)

is an isomorphism.

Since the inclusion is clearly injective, it suffices to show that it is surjective. IfX ∈ Vq,1 and z is a primitive q-th root of unity, we let

Xz(t) =1q

q−1∑j=0

z−jX(t+ 2πj).

Then

Xz(t+ 2π) =1q

q−1∑j=0

z−jX(t+ 2π(j + 1))

=z

q

q−1∑j=0

z−(j+1)X(t+ 2π(j + 1)) = zXz(t),

93

so Xz ∈ V1,z. Moreover,X =

∑zq=1

Xz,

so the inclusion (2.26) is indeed an isomorphism, proving the Lemma.

Note that if X ∈ V1,z1 and Y ∈ V1,z2 , where z1 and z2 are q-th roots of unity,then

Iq(X, Y ) =∫ 2π

0

[⟨DX

∂t,DY

∂t

⟩− 〈R(X, γ′)γ′, Y 〉

]dt

+∫ 4π

2π

z1z2

[⟨DX

∂t,DY

∂t

⟩− 〈R(X, γ′)γ′, Y 〉

]dt

+ · · ·+∫ 2πq

2π(q−1)

zq−11 zq−1

2

[⟨DX

∂t,DY

∂t

⟩− 〈R(X, γ′)γ′, Y 〉

]dt

=

q−1∑j=0

zj1zj2

∫ 2π

0

[⟨DX

∂t,DY

∂t

⟩− 〈R(X, γ′)γ′, Y 〉

]dt.

Thus we see that

Iq(X, Y ) =

qI1(XY ), if z1 = z2,

0, if z1 6= z2,

and hence the direct sum decomposition∑zq=1 V1,z of Vq,1 is orthogonal with

respect to the index form Iq. Let N(z) denote the nullity of the index form I1restricted to V1,z,

N(z) = dimCX ∈ V1,z : I1(X, Y ) = 0 for all Y ∈ V1,z.

The above discussion proves the following lemma due to Bott [8], which playsa key role in his analysis of the relationship between the index and nullity of aprime smooth closed geodesic and the index of nullity of its multiple covers:

Lemma 2.8.4. Let γq denote the q-fold iterate of the smooth closed geodesicsγ, so γq(t) = γ(qt). Then

Nullity of γq =∑zq=1

N(z).

We now return to the proof of the bumpy metric theorem itself. It suffices toconsider geodesics whose length is less than a given bound. We know that thereare only finitely many prime geodesics with length below this bound and that ifthe metric is perturbed by a sufficiently small amount, no new prime geodesicswill be introduced. Our strategy is to perturb the metric in a neighborhood ofa given geodesic in such a way that the geodesic is preserved.

As in the proof of the preceding Theorem, we construct a perturbation ofthe Riemannian metric on M of a specific form. Once again, we choose a point

94

t ∈ S1 and a neighborhood U containing t such that γ imbeds U into some openset W ⊂ M on which Fermi coordinates (x1, . . . , xn) are defined, coordinatessuch that:

1. γ is described by the equations x2 = · · · = xn = 0,

2. x1 γ = t,

3. the metric g takes the form∑gijdu

iduj , such that on γ(U), gij = δij , for1 ≤ i, j ≤ n.

Following Klingenberg ([42], Proposition 3.3.7), we construct a perturbation ofthe metric g =

∑gijdxidxj such that

g11(x1, . . . xn) =n∑

r,s=2

xrxsαrs(x1, x2, . . . , xn), gij = 0 if (i, j) 6= (1, 1).

Here αrs are smooth functions which vanish outside a small tubular neighbor-hood of γ. A straightforward calculation shows that the resulting changes inthe Christoffel symbols

Γkij =12

(∂gik∂uj

+∂gik∂uj

− ∂gik∂uj

)vanish unless at least two of the indices are 1, and if 2 ≤ r, s ≤ n,

Γr11 = −∑s

usαrs, Γ1r1 =∑s

usαrs.

The corresponding changes in curvature components are given by the formulae

Rlijk =∂

∂ui

(Γljk

)− ∂

∂uj

(Γlik

)+∑m

ΓlimΓmjk +∑m

ΓlimΓmjk −∑m

ΓljmΓmik −∑m

ΓljmΓmik.

But along f0(Σ0) all the Γkij ’s and Γkij ’s must vanish, so the resulting changein the curvature R1r1s along γ is given by

R1r1s(x1, 0, . . . , 0) = αrs(x1, 0, . . . , 0).

The Jacobi operator Lγ on normal sections can be expressed in componentsas follows: If

X =n∑r=2

fr∂

∂xr, then Lγ(X) = −

n∑r=2

[d2frdx2

1

+n∑s=2

R1r1sfs

]∂

∂xr.

We can consider the family of formally self-adjoint second order differentialoperators T of the form

T (X) = −n∑r=2

[d2frdx2

1

+n∑s=2

Trsfs

]∂

∂xr.

95

For each such operator and each q-th root of unity z ∈ S1 ⊂ C, we can definethe nullity

N(z) = dimCX ∈ V1,z : L(X) = 0.

For an open dense set of such operators T , N(z) = 0. The argument in thepreceding paragraph shows that we can perturb the Jacobi equation along anygeodesic so that N(z) = 0. Since there are a finite number of roots of unitycorresponding to covers γq of a fixed prime geodesic which have length less thana given bound, we can ensure that for generic metrics all such geodesics γq willbe Morse nondegenerate. This finally proves the Bumpy Metric Theorem forgeodesics.

2.9 Adding handles

Suppose that M is a smooth manifold modeled on a Hilbert space with a com-plete Riemannian metric, and that f :M→ R is a C2 Morse function satisfyingcondition C. From the Deformation Theorem 1.11.1, we know that the topologyof

Ma = p ∈M : f(p) ≤ a

does not change as a increases unless a passes a critical value for f . Our goalnow is to understand what happens to the topology of Ma when a passes acritical value c for f such that f−1(c) contains a finite number of critical points,all Morse nondegenerate. In this section, we carry our that analysis for the caseof Hilbert manifolds.

Theorem 2.9.1. Suppose that M is a Hilbert manifold with a complete Rie-mannian metric 〈〈·, ·〉〉 and that f :M→ [0,∞) is a smooth function satisfyingcondition C. If the interval [a, b] contains a single critical value c for f , thereis exactly one critical point p for f such that f(p) = c and this critical pointis Morse nondegenerate of Morse index λ, then Mb is homotopy equivalent toMa with a handle of index λ attached.

To prove this, we choose a coordinate chart (U, φ) about the nondegeneratecritical point p with φ(p) = 0 ∈ E, where E is the model space for M, and letf = f φ−1, a smooth function on the open subset φ(U) ⊂ E which has a singlenondegenerate critical point of index λ at the origin. Expanding the smoothfunction f in a Taylor series yields

f(x) = c+12d2f(0)(x, x) +R1(x), where ‖R1(x)‖ ≤ ε1‖x‖2, (2.27)

for x ∈ φ(U), where ε1 is a positive constant which can be made arbitrarilysmall by making U small. We can identify TpM with E and since p is Morsenondegenerate, the self-adjoint bounded linear operator A on E determined by

d2f(0)(x, y) = 〈〈Ax, y〉〉, for x, y ∈ E,

96

where 〈〈·, ·〉〉 is the Riemannian metric, is invertible, so its spectrum is a closedsubset of the real axis, bounded away from zero. Corresponding to the restric-tion of the vector field X = grad(f) to U is a vector field on φ(U) with localrepresentative X which has the Taylor series expansion

X (x) = Ax+R2(x), where ‖R2(x)‖ ≤ ε2‖x‖, (2.28)

for x ∈ φ(U), where ε2 is an arbitrarily positive constant. The vector field X onφ(U) is closely approximated by its linearization XA which has local representa-tive XA(x) = Ax, and one can check that this linearization is a pseudo-gradient(as defined in § 1.11) when ε2 is sufficiently small.

If the critical point p has finite index, the negative part of the spectrum isdiscrete and consists of finitely many eigenvalues. In this case, we let E− bethe subspace of E generated by the negative eigenvalues of A and let E+ bethe closed orthogonal complement to E− in E. In general, the spectral theoremallows us to divide E into a direct sum E = E+ ⊕ E−, each summand beingpreserved by A, and hence we can think of A as dividing into a “block matrix”

A =(A− 00 A+

)with A− and A+ self-adjoint invertible operators on the subspaces E− andE+ respectively. The spectrum of A− lies on the negative real axis, while thespectrum of A+ lies on the positive real axis. Thus if we suppose that x− and x+

are the orthogonal projections of x into E− and E+ respectively, so x = x−+x+,the system of differential equations represented by the linearization −XA is

dx−/dt = −A−x−,dx+/dt = −A−x+.

This is just a linear system with constant coefficients, which has φt : t ∈ R asits one parameter group of diffeomorphisms, where

φt(x−, x+) = (exp(−tA−x−), exp(−tA+x+)). (2.29)

The fluid flow for −XA described by φt : t ∈ R is expanding on the subspaceE−, contracting on E+ and closely approximates the fluid flow for X .

Let V be an open subset of U such that p ∈ V ⊆ V ⊆ U .

Lemma 2.9.2. There exist r1 > 0 and s1 > 0 such that

1. D−(r)×D+(s) ⊆ V and XA is transverse to D−(r)×∂D+(s) when r ≤ r1

and s ≤ s1,

2. f(∂D−(r1)×D+(s1)) ≤ c− ε, for some ε > 0, and

3. f−1(c− ε) is transverse to D−(r1)× x+ for x+ ∈ D+(s1).

97

The first condition follows immediately from the explicit form (2.29) of the fluidflow for XA.

To prove the second claim, we use the Taylor expansion (2.27) to concludethat

f(x) = c+12〈〈A−x−, x−〉〉+

12〈〈A+x+, x+〉〉+R1(x)

≤ c− 12‖A−1

− ‖‖x−‖2 + +

12‖A+‖‖x+‖2 + ε1‖x‖2.

We can choose ε1 smaller than (1/(4‖A−‖2). Thus if we choose r1 small andthen s1 much smaller, we can arrange that the second claim holds. The thirdclaim is proven in a similar fashion, and we leave it as an exercise for the reader.

In accordance with the terminology of geometric topology, we call ∂D−(r1)and D−(r1) the descending sphere and descending disk of the critical point p;∂D−(r1) is a sphere of dimension λ − 1. We also have an ascending sphere∂D+(s1) and an ascending disk D+(s1), both of which are infinite-dimensional.

Returning now to the proof of the theorem, we let η :M→ [0, 1] be a smoothfunction such that η(q) = 1 for q ∈M− U , η(q) = 0 for q ∈ V and let

X = η grad(f) + (1− η) φ−1∗ XA.

Finally, we let φt : t ∈ R be the one-parameter group of local diffeomorphismscorresponding to the vector field −X . For q ∈Mb −Mc−ε, let τ(q) denote thefirst time t such that

φt(q) ∈Mc−ε ∪ φ−1(D−(r1)× ∂D+(s1)).

Note that τ(q) is finite by the argument for the Theorem 1.10.1 and the transver-sality conditions of Lemma 2.8.4 show that τ(q) depends continuously on q. Letht :Mb →Mb by ht(q) = φtτ(x)(q). Then clearly h0 is the identity map and

h1(Mb) ⊂Mc−ε ∪ φ−1(D−(r1)× ∂D+(s1)).

In fact, we easily check that ht gives a deformation retraction from Mb toMc−ε ∪ φ−1(D−(r1) × ∂D+(s1)), which is homotopy equivalent to Mc−ε witha handle of index λ attached. By the Deformation Theorem 1.11.1 this has thehomotopy type of Ma with a handle of index λ attached.

Remark 2.9.3. In the proof, the Riemannian metric on M is used only toconstruct the continuous isomorphism A : TpM→ TpM and the gradient vectorfield X away from the critical point p. This suggests that it might be convenientto formulate the notion of “gradient-like” vector field, more flexible than anordinary gradient, a concept utilized by Milnor for finite-dimensional manifoldsin [51]:

Definition. Suppose that f : M → R is a C2 Morse function on a HilbertmanifoldM with a complete Riemannian metric, and let K ⊆M be the critical

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locus of f . We say that a C2 vector field X on M is gradient-like for f if Mpossesses an open cover U, V such that

1. the restriction of f to M−K is a pseudogradient for f ,

2. the open set U divides into connected components Up, one for each p ∈ K,such that Up is the domain of a coordinate system φp such that φp(p) = 0,and

3. (φp)∗(X|Up) = XA, where XA has local representative

XA(x) = Ax, for x ∈ φp(U), (2.30)

and A : TpM→ TpM is invertible and self-adjoint with respect to someHilbert space inner product 〈〈·, ·〉〉 on TpM.

If ηV ∪ ηp : p ∈ K is a Ck partition of unity with respect to the open coverV ∪ Up : p ∈ K, and XV is a pseudogradient for f on V , then

X = ηV XV +∑p∈K

ηp[(φ−1p )∗XA]

is a Ck gradient-like vector field for f onM, so long as the open neighborhoodsUp are chosen sufficiently small. Even if the Morse function f is only C2 (soits gradient is only C1), we can construct Ck gradient-like vector fields for f onM−K so long as M admits Ck partitions of unity.

Note that it was a gradient-like vector field for f that we utilized in theproof of Theorem 2.8.1. Since pseudogradients form an open subset of thespace of vector fields, gradient-like vector fields give considerable flexibility inconstructions involving flows corresponding to Morse functions, as we will seewhen constructing the Morse-Witten chain complex for a Morse function f inthe next section.

We can extend Theorem 2.9.1 to the case of several Morse nondegeneratecritical points:

Theorem 2.9.4. Suppose that M is a Hilbert manifold with a complete Rie-mannian metric 〈〈·, ·〉〉 and that f :M→ [0,∞) is a smooth function satisfyingcondition C. If the interval [a, b] contains a single critical value c for f and thereare finitely many critical point p1, . . . , pk for f such that f(pi) = c which areMorse nondegenerate with finite Morse indices λ1, . . . λk, thenMb is homotopyequivalent to Ma with handles of index λ1, . . . , λk attached.

The proof is a straightforward extension of the proof of Theorem 2.8.1.

Remark 2.9.5. Theorems 2.9.1 and 2.9.4 both hold for critical points of infi-nite Morse index, although attaching handles of infinite index turns out to beinvisible from the homotopy theory point of view. Moreover, in the problemswe have been considering from the calculus of variations, the Morse index of

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critical points is always finite, so the handles constructed via these theoremsare finite-dimensional.

The Bumpy Metric Theorem 2.8.2 provides motivation for extending Theo-rems 2.9.1 and 2.9.4 to the case of nondegenerate critical submanifolds. IfN ⊂ M is a compact nondegenerate critical submanifold of finite Morse indexλ, the normal bundle νN to N is defined via the Riemannian metric 〈〈·, ·〉〉 onM; its fiber at p ∈ N is

(νN)p = V ∈ TpM : 〈〈V,W 〉〉 = 0 for all W ∈ TpN.

The normal bundle has a finite-dimensional subbundle ν−N whose fiber at pis generated by eigenvectors corresponding to the negative eigenvalues of A :TpM→ TpM, the rank of the bundle ν−N being the Morse index of N . Let

D(ν−N) = V ∈ ν−N : ‖V ‖ ≤ 1, S(ν−N) = V ∈ ν−N : ‖V ‖ = 1,

the unit disk and unit sphere bundles in the negative normal bundle ν−N .

Theorem 2.9.6. Suppose that M is a Hilbert manifold with a complete Rie-mannian metric 〈〈·, ·〉〉 and that f :M→ [0,∞) is a smooth function satisfyingcondition C. If [a, b] contains a single critical value c for f , the set of criticalof points with value c forming a nondegenerate critical submanifold N of finiteMorse index λ, then Mb is homotopy equivalent to Ma with the disk bundleD(ν−N) attached to Ma along S(ν−N).

The proof is a relatively straightforward modification of the proof for Theo-rem 2.9.1. Note that we can allow N to have more than one component.

2.10 Morse inequalities

Suppose now thatM is a Hilbert manifold with a complete Riemannian metric〈〈·, ·〉〉 and that f :M→ [0,∞) is a Morse function satisfying condition C, allcritical points having finite Morse index. For a ∈ R, we let

Ma = p ∈M : f(p) ≤ a.

Condition C implies that eachMa has only finitely many nondegenerate criticalpoints. We would like to investigate how the topology of Ωa changes as aincreases. There are two ways of tracking the changes in topology, one is viathe Morse inequalities which we discuss next, the other via the Morse-Wittenchain complex for f which will be treated in § 2.11.

For each nonnegative integer λ, we let µaλ denote the number of criticalpoints for f withinMa of Morse index λ and for a given choice of field F (suchas R or Z2), we let βaλ be the dimension of Hi(Ma;F ). Then the weak Morseinequalities state that

µaλ ≥ βaλ. (2.31)

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Instead of proving these directly, we will prove a stronger version of the Morseinequalities in terms of the Morse polynomial for f and Poincare polynomial forMa, defined respectively by

Ma(t) =∞∑λ=0

µaλtλ, Pa(t) =

∞∑λ=1

βaλtλ.

Then (2.31) is an immediate consequence of:

Theorem 2.10.1. (Morse inequalities) M is a Hilbert manifold with acomplete Riemannian metric and that f : M → [0,∞) is a Morse functionsatisfying condition C. Then there is a polynomial

Qa(t) =∞∑λ=0

qaλtλ with qaλ ≥ 0 such that Ma(t)− Pa(t) = (t+ 1)Qa(t).

(2.32)

We make the simplifying assumption that there is only critical point for eachcritical value, so that we can apply Theorem 2.9.1 instead of Theorem 2.9.4.This can be arranged quite easily by simply perturbing f . Our strategy is toapply an induction on a.

To start the induction, we note that when we set a < 0, Ma is empty andhence βaλ = µaλ = 0. We can therefore set Qa(t) = 0 in this case.

For the inductive step, observe first that if the interval [a, b] contains a singlecritical value c corresponding to a Morse nondegenerate critical point of Morseindex λ, then by Theorem 2.9.1,

Hk(Mb,Ma;F ) ∼=

F, if k = λ,0, if k 6= λ.

Thus it follows from the exact sequence

· · · → Hk(Ma; Z)→ Hk(Mb; Z)→ Hk(Mb,Ma; Z)→ · · ·

that eitherPb(t) = Pa(t) + tλ or Pb(t) = Pa(t)− tλ−1,

and one can check that the two cases depend on whether the descending sphere(pushed down into Ma) bounds or not. In the former case, the descendingdisk can be completed to a cycle representing a new generator of λ-dimensionalhomology, while in the latter it yields a new relation in (λ − 1)-dimensionalhomology. Since Mb(t) =Ma(t) + tλ in either case, we see that

(Mb(t)− Pb(t))− (Ma(t)− Pa(t))

is either 0 or tλ−1(t+ 1). Thus assuming (2.32) for a, we can arrange that

Mb(t) = Pb(t) + (t+ 1)Qb(t)

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p1

p2

qf

Æ

Figure 2.1: If a smooth proper Morse function f : R2 → [0,∞) has two localminima at p1 and p2, it must have an additional critical point (say at q) ofMorse index one. This consequence of the Morse inequalities is known as the“mountain pass lemma.”

also holds by setting Qb(t) = Qa(t) or Qb(t) = Qa(t) + tλ−1. This finishes theinductive step, and the theorem follows.

If M has only finitely many critical points of each index, we can let µλ denotethe number of critical points for f of Morse index λ and let βλ be the dimensionof Hi(M;F ). If we define the Morse series for f and Poincare series forM by

M(t) =∞∑λ=0

µλtλ, P(t) =

∞∑λ=0

βλtλ,

it follows from Theorem 2.9.1 that there is a polynomial

Q(t) =∞∑λ=0

qλtλ with qλ ≥ 0 such that M(t)− P(t) = (t+ 1)Q(t). (2.33)

This is a statement of the Morse inequalities for f :M→ [0,∞).

Corollary 2.10.2. (Lacunary Principle) If µaλ 6= 0 ⇒ µaλ−1 = 0 = µaλ+1,then µaλ = βaλ, for every nonnegative integer λ.

Indeed, by the weak Morse inequalities, βaλ−1 = 0 = βaλ+1, and hence thecoefficients of λ − 1 and λ + 1 in (t + 1)Qa(t) must be zero. From this weconclude that Qa(t) ≡ 0.

Similarly, we can let a → ∞ and obtain the lacunary principle for f on M: Ifµλ 6= 0⇒ µλ−1 = 0 = µλ+1, then µλ = βλ, for every nonnegative integer λ.

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Example 2.10.3. Suppose that M = Sn, the n-dimensional sphere with metricof constant curvature one, where n ≥ 3, and that p and q are points in M whichare not antipodal. Then there is exactly one geodesic from p to 1 of Morse indexk(n− 1), for each k ∈ N ∪ 0, and hence

M(t) = 1 + tn−1 + t2(n−1) + · · ·+ tk(n−1) + · · · .

From the “feedback” implicit in the Morse inequalities (2.33), we see thatM(t) = P(t), so

P(t) = 1 + tn−1 + t2(n−1) + · · ·+ tk(n−1) + · · · ,

and we conclude that for any field F ,

Hm(Ω(Sn, p, q);F ) ∼=

F, if m = k(n− 1) or k ∈ N ∪ 0,0, otherwise.

Thus if 〈·, ·〉 is any Riemannian metric on Sn, then for generic choice of p and qin Sn, there will be infinitely many geodesics from p to q, at least one being ofindex k(n− 1) for each k ∈ N ∪ 0.

We would also like Morse inequalities for the case whereM = L21(S1,M), where

M is a compact Riemannian manifold. If we give M a “generic” Riemannianmetric, Theorem 2.7.5 implies that all nonconstant critical points for the actionJ : L2

1(S1,M)→ R lie on one-dimensional nondegenerate critical submanifolds.We can suppose that the metric is chosen so that only one O(2)-orbit of criticalpoints lies at each critical level.

Thus suppose that the interval [a, b] contains a unique critical value c withJ−1(c) containing a unique O(2)-orbit of geodesics which comprise a nonde-generate critical submanifold N . Then Theorem 2.8.6 gives an isomorphism oncohomology

Hk(Mb,Ma; Z) ∼= Hk(D(ν−N), S(ν−N); Z).

It follows from the Thom isomorphism theorem (with twisted coefficients) thatif the Morse index of N is λ, then

Hk(D(ν−N), S(ν−N); Z) ∼= Hk−λ(N ; Z⊗ θ−),

where θ− is the orientation bundle of N . If the bundle ν−N is orientable, twistedhomology reduces to usual homology, and hence

Hk(D(ν−N), S(ν−N); Z) ∼= Hk−λ(N ; Z).

If N is a nondegenerate critical submanifold for J , we let

PN (t) =∑λ

[dimHλ(N ; θ− ⊗ F )]tλ.

In the case where the normal bundle is orientable, this is the Poincare polynomialof N . If F = R and N is a nondegenerate O(2)-orbit of geodesics, then

PN (t) = 2(1 + t) or PN (t) = 0,

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depending upon whether the normal bundle is orientable or not, while if F = Z2,we always have PN (t) = 2(1 + t). The Morse polynomial of J on Ma is

Ma(t) =∑tλNPN (t) : N is a nondegenerate critical submanifold

of Morse index λN with J(N) ≤ a .

As in the case of a Morse function, the Morse series is related to the Poincarepolynomial of Ma,

Pa(t) =∞∑λ=0

dimHλ(Ma;F )tλ,

by the Morse inequalities, which state that there is a polynomial

Qa(t) =∞∑i=1

qaλtλ with qλ ≥ 0 such that Ma(t) = Pa(t) + (t+ 1)Qa(t).

(2.34)If we know the homology (or cohomology) of L2

1(S1,M)a, the Morse inequalities(2.34) enable us to estimate the number of smooth closed geodesics M must havewith action ≤ a when M is given a generic metric.

2.11 The Morse-Witten complex

The Morse inequalities do not usually completely determine the integer homol-ogy H∗(Mb; Z). The additional information we would need for an inductivedetermination of H∗(Mb; Z) is the boundary map in the long exact sequence

· · · → H∗(Ma; Z)→ H∗(Mb; Z)→ H∗(Mb,Ma; Z)→ · · · .

However, implicit in the writings of Thom, Smale, Milnor and others, is a proce-dure for calculating the boundary map by determining the trajectories betweencritical points. This results in a chain complex (C∗(f,X ), ∂) that depends onthe function f and on a gradient-like vector field X used to calculate trajectoriesbetween critical points, a chain complex which calculates the homology of themanifold Mb. This chain complex is often called the Morse-Witten chain com-plex , because of the fact that Witten gave an important quantum-mechanicalinterpretation of the boundary operator [83]. We merely sketch the ideas of theconstruction here; a much more complete treatment can be found in [74] andChapter 6 of [40].

For the statement of the next theorem, we assume the reader is familiar withthe notion of CW complex; background on this topic can be found in Chapter 0of [35].

Theorem 2.11.1. If f :M→ [0,∞) is a Morse function on a complete Hilbertmanifold that satisfies condition C and all of its critical points have finite index,then for each a ∈ R, Ma has the homotopy type of a finite CW complex withone cell of dimension λ for each critical point of index λ.

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Sketch of proof: This is a consequence of Theorem 2.9.4 and it is an analog ofTheorem 3.5 in [50]. In fact, one could prove that M itself has the homotopytype of a CW complex, but a rigorous proof would require a limiting procedureas a→∞. Such a procedure is given in Appendix A of [50].

Thus the homology of M should be just the cellular homology of the resultingCW complex, as described in books on algebraic topology; see, for example,[35], pages 137-141. This raises the problem of describing the cells and attachingmaps, and finding an algorithm for calculating this homology.

Such an algorithm is based upon choice of a gradient-like vector field X forthe Morse function f : M → R. Let φt : t ∈ R be the one-parameter groupof local diffeomorphisms corresponding to −X .

Definition. The unstable manifold Wp(f,X ) of a critical point p ∈M consistsof the images of all trajectories t 7→ φt(q) which start at p in the remote past,in other words, such that φt(q)→ p as t→ −∞. Similarly, the stable manifoldW ∗p (f,X ) of a critical point p consists of the images of trajectories t 7→ φt(q)such that φt(q)→ p as t→∞.

Lemma 2.11.2. The unstable and stable manifolds Wp(f,X ) and W ∗p (f,X )are in fact submanifolds of M.

Sketch of proof: Let us consider the case of the unstable manifold Wp(f,X ).First one uses the explicit description (2.30) of the gradient-like vector field Xnear p to show that for ε > 0 sufficiently small,

Wp(f,X )ε = φ−1p (x ∈ E− : ‖x‖ < ε)

is part of the unstable manifold. Then one notes that

Wp(f,X ) =⋃φt(Wp(f,X )ε) : t ≥ 0.

The properties of smooth flows corresponding to vector fields then show thatWp(f,X ) is a smooth manifold diffeomorphic to an open cell. The proof forW ∗p (f,X ) is similar, starting with

W ∗p (f,X )ε = φ−1p (x ∈ E+ : ‖x‖ < ε) .

It follows from Condition C and the explicit form of X that any orbit q 7→ φt(q)converges to some critical point as t→∞ (although it may come close to severalother critical points first).

Lemma 2.11.3. We can adjust the gradient-like vector field for f so that if λpis the index of p and λq is the index of q,

1. λp ≤ λq ⇒Wp(X ) ∩W ∗q (X ) is empty, while

2. λp > λq ⇒Wp(X ) ∩W ∗q (X ) is a submanifold of dimension λp − λq, whenX is sufficiently smooth.

105

Sketch of proof: If Wp(X ) ∩W ∗q (X ) is nonempty, then f(p) > f(q) and we canchoose a regular value c for f such that f(p) > c > f(q). Then N = f−1(c) isa codimension one submanifold of M, and

dim (N ∩Wp(X )) = λp − 1, codim (N ∩W ∗q (X )) = λq.

Moreover, if we choose c sufficiently close to the Morse nondegenerate criticalpoint p, we see that S = N ∩Wp(X ) is a (λp − 1)-dimensional sphere and liesin an open tubular neighborhood U of N with diffeomorphism ψ : U → S × V ,where V is an open ball in a Banach space, with ψ(S) = S × 0. Let

N = U ∩W ∗q (X ), and consider g = π ψ : N → V,

where π is the projection on the second factor. We can check that g is a Fredholmmap with Fredholm index λp − 1− λq, and

ψ−1(S × x) ∩N = s ∈ N : g(s) = x.

Assuming that X (and hence N) is sufficiently smooth, we can use the Sard-Smale Theorem 2.6.2 to choose a regular value x for g. Finally, we can choosex as close as we want to 0, and construct an isotopy from a neighborhood ofS to itself which carries S × 0 to S × x. Since the conditions definingpseudogradient are open, we can replace X by a new gradient-like vector fieldso that

N ∩Wp(X ) and N ∩W ∗q (X )

have transverse intersection. If λp ≤ λq, the dimension of N ∩Wp(X ) is lessthan the codimension of N ∩W ∗q (X ), so Wp(X )∩W ∗q (X ) is empty. If λp > λq,then N ∩Wp(X ) ∩W ∗q (X ) is a submanifold of dimension λp − λq − 1.

In particular, if λp = λq + 1, then N ∩ Wp(X ) ∩ W ∗q (X ) consists of a finitenumber of points and Wp(X ) ∩W ∗q (X ) consists of a finite collection of smoothcurves from p to q.

We can now orient the unstable manifolds and define the Morse-Wittencomplex of the nonnegative Morse function f . We let Cλ+1(f,X ) be the freeZ-module generated by the critical points pλ+1,1, pλ+1,2 . . . of f of index k andlet ∂ be the Z-module homomorphism

∂ : Cλ+1(f,X ) −→ Cλ(f,X )

defined by∂(pλ+1,j) =

∑q

ajqpλ,q, (2.35)

where ajq ∈ Z is the oriented number of trajectories from pλ+1,j to pλ,q. Thesign of a trajectory γ : (−∞,∞) → M from p to q is determined as follows:First one orients each unstable manifold for f . Then one constructs a trivialvector bundle E along γ which is transverse to TW ∗q (X ) and of complementarydimension such that the fiber Eγ(t) approaches a hyperplane in the negative

106

eigenspace TpM− for d2f(p) as t→ −∞, and the positive eigenspace for d2f(q)as t → ∞. Using this bundle we translate the oriented negative eigenspaceTqWq(X ) for d2f(q) back to p. If TqM− denotes the resulting hyperplane inTpWp(X ), then

TpWp(X ) = TqM− ⊕ T,

where T is the oriented line in the direction of the trajectory leaving p. Weassign the positive sign to the trajectory γ if the orientation of TpWp(X ) agreeswith the direct sum orientation, the minus sign otherwise.

The next theorem says that (C∗(f,X ), ∂) is a chain complex:

Theorem 2.11.4. Suppose that f : M → R is a nonnegative Morse functionon a complete Hilbert manifold that satisfies condition C and all of the criticalpoints of f have finite index. Then ∂ ∂ = 0 and the homology (with integer co-efficients) of the Morse-Witten complex (C∗(f,X ), ∂) is just the usual homologyH∗(M; Z).

Sketch of proof: It suffices to prove this for the restriction of f to Ma whichhas only finitely many critical points, and we henceforth use the notationM forMa. After generic choice of X , follows from Lemma 2.10.3 that unstable criticalpoints of index λ intersect only stable manifolds of critical points of index < λ.Thus if we let M(λ) denote the union of the unstable manifolds in M of index≤ λ, we obtain an increasing filtration

· · · ⊂ M(λ−1) ⊂M(λ) ⊂M(λ+1) ⊂ · · ·

of M by closed subsets. Since the unstable manifolds of index λ are in one-to-one correspondence with generators of

H∗(M(λ),M(λ−1); Z),

we can set C∗(f,X ) equal to this homology group. If we let ∂′ denote theboundary homomorphism in the exact sequence of the triple

(M(λ),M(λ−1),M(λ−2)),

one can use the usual argument from the theory of cellular homology to showthat ∂′ ∂′ = 0 and the cohomology of the Morse complex is the standardhomology of M (as in the proof of Theorem 2.35 in [35]). Thus it is relativelyeasy to see that we do indeed get a chain complex, and we have reduced theproof to the verification that ∂′ is the same as the homomorphism ∂ defined by(3.44).

For this, one can follow arguments used by Milnor ([51]) to prove the h-cobordism theorem of Smale. Note we can adjust the values of the Morse func-tion f while simultaneously multiplying the gradient-like vector field X by apositive function η : Ma → (0,∞). In this way, we can replace the Morsefunction f by “self-indexing” function f∗ which satisfies the conditions f? ≥ −1and f?(p) = λ, whenever p is a critical point of index λ, by following the proof

107

of Theorem 4.8 in [51], without changing critical points or the boundary mapsin the Morse-Witten complex. After this is done, M(λ) is a strong deformationretract of

Mλ+1/2 =q ∈M : f∗(q) ≤ λ+

12

.

To complete the sketch of the proof, we need only verify the following lemma:

Lemma 2.11.5. The Witten boundary ∂ agrees with the boundary ∂′ definedby the CW decomposition.

The proof is a straightforward modification of the proof of Theorem 7.4 in [51],to which we refer for details. Here we give only a very superficial description ofhow the argument goes. The key idea is that if

N = (f∗)−1(λ+ 1/2),

then the CW boundary ∂ can be regarded as a composition

Hλ+1(Mλ+3/2,Mλ+1/2; Z) −→ Hλ(N ; Z) −→ Hλ(Mλ+1/2,Mλ−1/2; Z)

where the first map takes the homology class corresponding to the (λ + 1)-handle for a critical point pλ+1,j to the homology class of its boundary sphereSλ+1,j ⊆ N , and the second map is induced by inclusion. On the cochain level,we have a corresponding factorization of the coboundary

Hλ(Mλ+1/2,Mλ−1/2; R) −→ Hλ(N ; R) −→ Hλ+1(Mλ+3/2,Mλ+1/2; R),

where we use real coefficients so that we can represent cycles by differentialforms. Corresponding to a critical point pλ,q we can define a “Thom form” θλ,q(as described in [10]) which represents a cohomology class

[θλ,q] ∈ Hλ(Mλ+1/2,Mλ−1/2; R)

such that ∫Wpλ,r

θλ,q = δrq =

1 if r = q,

0 if r 6= q.

We can think of θλ,q as Poincare dual to the class represented by the inifinite-dimensional stable manifold for pλ,q. One now checks that

ajq =∫Sλ+1,j

θλ,q ∈ Z

is both the integer appearing in the Witten boundary (3.44) and the integerappearing in the formula for the CW boundary.

Corollary 2.11.6. (Lacunary Principle) If µaλ 6= 0 ⇒ µaλ−1 = 0 = µaλ+1,then the boundary ∂ in the Morse-Witten chain complex is zero.

The proof is immediate.

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Note that applying Corollary 2.11.6 to Example 2.10.3. yields the integer ho-mology of the loop space of Sn when n ≥ 3:

Hm(Ω(Sn, p, q); Z) ∼=

Z, if m = k(n− 1) or k ∈ N ∪ 0,0, otherwise.

In spite of the fact that the Morse-Witten complex gives stronger results, theMorse inequalities are often quite useful, since in many cases it is difficult tocalculate the boundary operator ∂.

109

Chapter 3

Harmonic and minimalsurfaces

3.1 The energy of a smooth map

Suppose now that Σ is a compact smooth Riemannian manifold of dimension m.In terms of local coordinates (u1, . . . , um) on Σ we can write the Riemannianmetric and area or volume element on Σ as

m∑a,b=1

ηabdua ⊗ dub and dA =

√ηdu1 · · · dum,

where η denotes the determinant of the matrix (ηab). If M is a second Rieman-nian manifold of dimension n isometrically imbedded in Euclidean space RNand f : Σ→M ⊆ RN is a smooth map, the energy density of f at a given pointis given in terms of local coordinates by the formula

e(f) =12|df |2, where |df |2 =

m∑a,b=1

ηab∂f

∂ua· ∂f∂ub

,

where (ηab) denotes the matrix inverse to (ηab) and the dot denotes the Eu-clidean dot product in RN . The energy of a smooth map f : Σ → M ⊆ RN isgiven by the Dirichlet integral

E(f) =∫

Σ

e(f)dA =12

∫Σ

m∑a,b=1

ηab∂f

∂ua· ∂f∂ub√ηdu1 · · · dum, (3.1)

the integrand being independent of choice of local coordinates. Note that if Σis one-dimensional, the energy reduces to the action J that we studied in theprevious chapter.

The energy defines a smooth map

E : C2(Σ,M) −→ R.

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To find the critical points of E, we consider a variation of a given map f whichhas its support within a coordinate chart (U, (u1, . . . , um)) on Σ. Such a varia-tion is simply a smooth family of maps t 7→ f(t) in C2(Σ,M) such that f(0) = fand f(t)(p) = f(p) for all t, when p ∈ Σ−U . In terms of the local coordinates,we set

α(u1, . . . , um, t) = f(t)(u1, . . . , um),

and a straightforward calculation shows that

d

dt(E(f(t))

∣∣∣∣t=0

=∫

Σ

m∑a,b=1

√ηηab

∂2α

∂t∂ua· ∂α∂ub

du1 · · · dum∣∣∣∣∣∣t=0

= −∫

Σ

∂α

∂t·

m∑a,b=1

∂

∂ua

(√ηηab

∂α

∂ub

) du1 · · · dum∣∣∣∣∣∣t=0

.

We can evaluate at t = 0, setting

V (u1, . . . um) =∂α

∂t(u1, . . . um, 0),

to obtain the first variation formula,

dE(f)(V ) = −∫

Σ

V ·

1√η

m∑a,b=1

∂

∂ua

(√ηηab

∂f

∂ub

) dA. (3.2)

If f is a critical point for the energy E, then dE(f)(V ) = 0 for all such variationsV , and f must satisfy the partial differential equation m∑

a,b=1

1√η

∂

∂ua

(√ηηab

∂f

∂ub

)> = 0, (3.3)

where (·)> denotes projection into the tangent space to M . Maps f ∈ C2(Σ,M)which satisfy equation (3.3) are called harmonic maps. Just like the equation forgeodesics, the equation for harmonic maps is nonlinear because of the projectioninto the tangent space. In fact, we can rewrite (3.3) in terms of the Levi-Civitaconnection D on M as

1√η

m∑a,b=1

D

∂ua

(√ηηab

∂f

∂ub

)= 0,

or alternatively, in terms of local coordinates (x, . . . xn) on M , we can write theequation of harmonic maps as

1√η

m∑a,b=1

∂

∂ua

(√ηηab

∂xi

∂ub

)+∑

Γijkηab ∂x

j

∂ua∂xk

∂ub= 0,

111

where the Γijk’s are the Christoffel symbols.Harmonic maps between Riemannian manifolds were introduced by Eells

and Sampson [18] in 1964, who used the heat flow approach to establish thefollowing theorem:

Theorem 3.1.1 (Eells and Sampson). If M is a compact connected Rie-mannian manifold with nonpositive sectional curvatures and Σ is a compactRiemannian manifold, then every component of C2(Σ,M) contains a minimalenergy representative, which is a smooth harmonic map.

A very nice proof of this theorem using heat flow can be found at the beginningof Chapter 5 of [45]. A fews years after Eells and Sampson, Hartman [34]established a uniqueness result:

Theorem 3.1.2 (Hartman). If Σ and M are compact connected Riemannianmanifolds and M has negative sectional curvatures, then any component ofC2(Σ,M) contains at most one harmonic map.

One should note, however, that if M has nonpositive sectional curvature, theHadamard-Cartan Theorem asserts that the exponential map expp : TpM →Mis a smooth covering at any point p ∈ M , so M has Euclidean space as itsuniversal cover, and all of its higher homotopy groups are zero. In this case, Mis a K(π, 1) and its topology is relatively simple.

The Eells-Sampson Theorem does not hold without the curvature assump-tion. Indeed, it fails spectacularly when the dimension of Σ is ≥ 3, and attemptsto minimize energy within a given homotopy class of maps can lead to elementsof L2

1(Σ,M) which have quite bad singularities, as discussed for example in [45].Later we will prove the above theorems in the case where Σ has dimension

two, as part of a general theory that will give existence results even when wemake no assumptions on the curvature of the range M . In the case where Σ hasdimension two, the energy is conformally invariant, and the theory of harmonicmaps simplifies considerably due to the existence of isothermal coordinates onΣ. A nice proof of existence of isothermal charts, based upon Hodge theory, canbe found in Chapter 5, §10 of [79]. According to this existence theorem, any two-dimensional Riemannian manifold possesses an atlas (Uα, (uα, vα)) : α ∈ Aconsisting of isothermal charts, charts such that the Riemannian metric on Σtakes the form

ds2 = λ2α(duα ⊗ duα + dvα ⊗ dvα),

where each λα is a positive smooth real-valued function on Uα. If ψα : α ∈ Ais a partition of unity subordinate to this atlas, then the energy of a C2 mapf : Σ→M is given by the formula

E(f) =12

∑α

∫Σ

ψα

[⟨∂f

∂uα,∂f

∂uα

⟩+⟨∂f

∂vα,∂f

∂vα

⟩]duαdvα, (3.4)

where 〈·, ·〉 is the Riemannian metric on M induced by the dot product on RN ,

112

and harmonic maps are simply the maps

f : Σ −→M such that(∂2f

∂u2α

+∂2f

∂v2α

)T= 0. (3.5)

If one takes two coherently oriented isothermal charts (Uα, (uα, vα)) and(Uβ , (uβ , vβ)), one finds by a short calculation that on the intersection Uα ∩Uβ ,

∂uα∂uβ

=∂vα∂vβ

,∂uα∂vβ

= −∂vα∂uβ

.

These are just the Cauchy-Riemann equations which state that wα = uα + ivαis a holomorphic function of wβ = uβ + ivβ . Thus if Σ is an oriented surface,the atlas (Uα, zα) : α ∈ A of positively oriented isothermal charts makes Σinto a one-dimensional complex manifold, otherwise known as a Riemann sur-face. In this way, Riemannian metrics on Σ are divided into equivalence classes,depending upon which Riemann surface structure is defined by the isothermalcharts.

It is often convenient to choose a canonical metric on Σ within the conformalequivalence class determined by a Riemann surface structure. The uniformiza-tion theorem from Riemann surface theory states that any compact Riemannsurface has as its universal cover one of three simply connected Riemann sur-faces,

S2 = C ∪ ∞, C or D = z ∈ C : |z| < 1.

The remarkable fact is that each of these model spaces possesses a Riemannianmetric of constant Gaussian curvature compatible with its conformal structure,and the deck transformations of the universal cover of the compact Riemannsurface Σ are isometries with respect to this Riemannian metric. If Σ is closed,we can normalize this metric by assuming that it has total area one, and it isthen unique up to diffeomorphism. It follows from the Gauss-Bonnet formulathat the sign of the curvature is positive, zero or negative when Σ has genuszero, one, or at least two, respectively.

In terms of the complex partial differential operators

∂

∂w=

12

(∂

∂u−√−1

∂

∂v

),

∂

∂w=

12

(∂

∂u+√−1

∂

∂v

)we can write (3.3) as

D

∂w

(∂f

∂w

)=(

∂2f

∂w∂w

)T= 0. (3.6)

Note that we can regard

∂f

∂was a section of E = f∗TM ⊗ C,

113

a complex vector bundle over the Riemann surface Σ. The vector bundle E hasa Hermitian metric defined by

v, w ∈ TpM ⊗ C 7→ 〈v, w〉,

where w is the conjugate of w, and the Levi-Civita connection D on f∗TMextends complex linearly to a metric connection on E. The following theoremstates that this connection gives E a canonical holomorphic structure, therebymaking possible a remarkable relationship between harmonic surfaces and com-plex analysis:

Theorem 3.1.3 (Koszul and Malgrange). If E is a complex vector bundlewith Hermitian metric over a Riemann surface Σ and D is a metric connectionon E, then there is a unique holomorphic structure on E such that if σ is asection of E,

σ is holomorphic ⇔ Dσ

∂w= 0,

whenever w is a complex coordinate on Σ.

We will not prove this theorem here. An extension of this theorem is proven asTheorem 5.1 in Atiyah, Hitchin and Singer [5] using the Newlander-Nirenbergtheorem on integrability of almost complex structures. Another proof is givenin Donaldson and Kronheimer [14], Theorem 2.1.53.

Thus we see that the equation for harmonic maps from oriented surfaces canbe expressed quite simply in terms of Riemann surface theory: Equation (3.6)asserts that a map f : Σ→M is harmonic if and only if in terms of any complexcoordinate w on Σ, the section

∂f

∂wof E = f∗TM ⊗ C

is holomorphic with respect to the holomorphic structure on E which is providedby the Koszul-Malgrange Theorem.

The locally defined holomorphic section

∂f

∂wmay have isolated singularities, but

[∂f

∂w

]: Σ→ P(E)

extends to all of Σ, where P(E) denotes the bundle of projective spaces of fibersof E = f∗TM ⊗ C. Under change of complex coordinate

∂f

∂w=∂f

∂z

∂z

∂w= (complex-valued function)

∂f

∂z,

and hence the various locally defined complex derivatives of f determine a com-plex line subbundle L of E, which is holomorphic by the Koszul-MalgrangeTheorem. The line bundle L is isomorphic to the tangent bundle to Σ if andonly if f has no branch points, in accordance with the following definition:

114

Definition. A point p ∈ Σ is a branch point for the harmonic map f : Σ→Mif (∂f/∂z)(p) = 0, where z is any complex coordinate near p.

If p is a branch point for f , and z is a local complex coordinate defined on asmall open neighborhood U of p with z(p) = 0, then we can write (∂f/∂z) = zνgfor some positive integer ν, where g is a section of L over U such that g(p) 6= 0.We call ν the order or multiplicity of the branch point. If we let w = zν+1, thendw = (ν + 1)zνdz and we can define a section

∂

∂wof L by

∂

∂w=

1(ν + 1)zν

∂

∂z.

Thus we see that the restriction of L to U is obtained from the holomorphictangent bundle TΣ|(U −p) by pasting it to a trivial bundle U ×C over U bymeans of the transition function

g0p =1

(ν + 1)zν: U − p) −→ C− 0).

From the trivial bundle over Σ − p) and this transition function one canconstruct the point bundle ζνp over Σ which has first Chern number c1(ζνp )([Σ]) =ν. If f has a single branch point p of branching order ν then L = TΣ⊗ ζνp .

In general, the divisor of the harmonic map f is the finite sum

(f) = ν1p1 + · · ·+ νnpm,

where p1, . . . , pm are the branch points of f and ν1, . . . , νm are their branchingorders. Then the above discussion shows that

L = TΣ⊗ ζν1p1 ⊗ · · · ⊗ ζνmpm .

If ν denotes the total branching order of f , the total number of branch points off , counted with multiplicity, then the first Chern number 〈c1(L), [Σ]〉 of the linebundle L (also known as the degree of L in many books on Riemann surfaces)is determined by the formula

〈c1(L), [Σ]〉 = 2− 2g + ν (3.7)

where g is the genus of Σ.Suppose, for example, that h : Σ2 →M is a smooth harmonic map without

branch points and that g : Σ1 → Σ2 is a nontrivial holomorphic branched cover.(Thus, if Σ2 = S2, the Riemann sphere, we can regard g as a meromorphicfunction on Σ1.) Then the composition f = h g : Σ1 → M is also harmonic;it is called a branched cover of the harmonic map h : Σ2 → M . We say thatthe harmonic map f is prime if it is not a nontrivial branched cover of anotherharmonic map. Note that if f : Σ→M is a k-fold branched cover of a harmonicsphere h : S2 →M which is free of branch points, then the line bundle L for fsatisfies

〈c1(L), [Σ]〉 = 2k.

115

3.2 Minimal surfaces

The theory of minimal surfaces in a Riemannian manifold M is concerned withcritical points of the area function

A : Map(Σ,M) −→ R defined by A(f) =∫

Σ

∣∣∣∣∂f∂u ∧ ∂f∂v∣∣∣∣ dudv,

which we will see is closely related to critical points for the energy function Edefined by (3.4). In contrast to E, the integrand for A is independent of thechoice of parametrization, and in particular does not depend upon the choice ofa metric or conformal structure.

Proposition 3.2.1. If Σ is given a conformal structure ω and f ∈ Map(Σ,M),then E(f) ≥ A(f), with equality holding if and only if f satisfies the conditions⟨

∂f

∂u,∂f

∂u

⟩=⟨∂f

∂v,∂f

∂v

⟩,

⟨∂f

∂u,∂f

∂v

⟩= 0, (3.8)

when w = u+ iv is any choice of complex chart, or equivalently,⟨∂f

∂w,∂f

∂w

⟩= 0, (3.9)

and 〈·, ·〉 denotes the Riemannian metric on the ambient manifold M .

Proof: We utilize two well-known algebraic identities for vectors v and w inRN :

|v ∧w|2 + |v ·w|2 = |v‖2|w|2, |v||w| ≤ 12(v|2 + |w|2

),

with equality holding only if |v| = |w|. Using these two facts, we find that∣∣∣∣∂f∂u ∧ ∂f∂v∣∣∣∣ ≤ ∣∣∣∣∂f∂u

∣∣∣∣ ∣∣∣∣∂f∂u∣∣∣∣ ≤ 1

2

(∣∣∣∣∂f∂u∣∣∣∣2 +

∣∣∣∣∂f∂v∣∣∣∣2), (3.10)

with equality holding if and only if (3.8) holds. This proves the proposition.

Definition. We say that a map f : Σ → M is weakly conformal with respectto a Riemann surface structure ω on Σ if it satisfies (3.9) when w = u+ iv is acomplex coordinate for ω.

Weak conformality of an harmonic map f : Σ→M is expressed by the vanishingof the Hopf differential , which is the quadratic differential

Ω(f) =⟨∂f

∂w,∂f

∂w

⟩dw ⊗ dw. (3.11)

It follows immediately from (3.6) that

∂

∂w

⟨∂f

∂w,∂f

∂w

⟩= 2

⟨D

∂w

∂f

∂w,∂f

∂w

⟩= 0,

116

so the Hopf differential of a harmonic map is indeed a holomorphic quadraticdifferential, as studied in Riemann surface theory.

Theorem 3.2.2. A weakly conformal harmonic map f : Σ → M is a criticalpoint for the area function A.

Proof: If f is weakly conformal and harmonic, and t 7→ f(t) is a smooth familyof maps with f(0) = f , then

E(f(t)) ≥ A(f(t)) and E(f(0)) = A(f(0))

⇒ d

dtE(f(t))

∣∣∣∣t=0

=d

dtA(f(t))

∣∣∣∣t=0

,

so f is critical for the area function.

Theorem 3.2.2 provides motivation for the following definition of parametrizedminimal surface:

Definition. A parametrized minimal surface is a weakly conformal harmonicmap f : Σ→M .

Note that if f is an immersion, we could give Σ the Riemannian metric whichmakes f an isometric immersion. This gives Σ a Riemann surface structurewhich makes f is weakly conformal, and with this conformal structure, f isharmonic if and only if it is minimal.

More generally, if f : Σ → M is a weakly conformal map, then the com-plex line bundle L we described in the preceding section is “isotropic,” that is,〈L,L〉 = 0. This constrains the possible singularities of f ; the differential dfp atany point p ∈ Σ has rank two if f is an immersion in a neighborhood of p, orrank zero if p is a branch point, but never has rank one.

We would like to consider minimal surfaces as critical points for a variationalproblem which involves the Dirichlet integral and the conformal structure, ratherthan the area function. In order to do this, we must allow the conformal struc-ture on Σ to vary, as well as the map f ∈ C2(Σ,M). In this and the followingsection, we suppose that Σ is a compact oriented connected surface withoutboundary.

The simplest case is that where Σ = S2, which has a unique conformalstructure by the uniformization theorem. We can regard Σ = S2 as C ∪ ∞and take the atlas defined by the standard coordinate z on C and the coordinatew = 1/z on S2 − 0. Then

∂f

∂z=∂f

∂w

∂w

∂z= − 1

z2

∂f

∂w,

and∂f

∂wbounded near ∞ ⇒ ∂f

∂z→ 0 like 1/z2 as z →∞. (3.12)

By the Koszul-Malgrange Theorem 3.1.3, we can regard ∂f/∂z as a holomorphicsection on S2−∞ with a removeable singularity at∞, and by the removeable

117

singularity theorem from complex analysis, ∂f/∂z extends to a holomorphicvector field on S2. Thus when calculating energy on the Riemann sphere, weare fully justified in using a single coordinate z = x+ iy on C = S2−∞, andthe energy can be expressed by the improper integral

E(f) =12

∫C

[⟨∂f

∂x,∂f

∂x

⟩+⟨∂f

∂y,∂f

∂y

⟩]dxdy.

Conversely, if f : C → M is a harmonic map of finite energy, in other words,such that the above integral is finite, a removeable singularity theorem of Sacksand Uhlenbeck (Theorem 3.6 in [68]) will show that f extends to a harmonicmap from S2 into M .

Proposition 3.2.3. A harmonic map f : S2 → M is automatically weaklyconformal, and hence a parametrized minimal surface.

To prove this, we let z = x + iy be the standard complex coordinate on C =S2 − ∞. Then it follows from (3.6) that

∂

∂z

⟨∂f

∂z,∂f

∂z

⟩= 2

⟨D

∂z

(∂f

∂z

),∂f

∂z

⟩= 0, so

⟨∂f

∂z,∂f

∂z

⟩is a holomorphic function which extends to a C2 function on S2 by (3.12).This function must be constant by the maximum modulus principle, and sinceit vanishes at ∞, the constant must be zero, establishing the assertion. Notethat the argument simply shows that the Hopf differential (3.11) is zero, so onecould prove the Proposition by simply citing the well-known fact from Riemannsurface theory that S2 has no nonzero holomorphic quadratic differentials.

We next consider the case where Σ = T 2 = S1 × S1, a torus. We imaginethat we have fixed a basis (α, β) for H1(T 2; Z), representing the two S1 factors.The covering transformations for the universal cover π : C → T 2 are invertibleholomorphic maps from C to itself, and it follows from the maximum modulusprinciple they must be translations. After performing a possible rotation ofC and a uniform stretching, we can arrange that one of the translations behorizontal with unit displacement. The other displacement can be representedby a point ω = u+ iv in the upper half-plane

H = ω = u+ iv ∈ C : v > 0.

Thus we can arrange that the conformal structures on T 2, for fixed a basis forH1(T 2; Z), are in one-to-one correspondence with points in the upper half-planeH, the point ω = u + iv ∈ H corresponding to the conformal class of the torusC/Λ, where Λ is the lattice in C generated by ω and 1, with α correspondingto ω and β corresponding to 1. We say that H is the Teichmuller space of thetorus and write T1 = H.

A change of basis for the lattice Λ is represented by multiplication by amatrix in SL(2,Z),(

αβ

)7→(a bc d

)(αβ

)for

(a bc d

)∈ SL(2,Z). (3.13)

118

The corresponding SL(2,Z)-action on the generators ω1 and ω2 of an arbitraryintegral lattice in the complex plane is(

ω1

ω2

)7→

(a bc d

)(ω1

ω2

)while the corresponding action on the coordinate ω = u+ iv = ω1/ω2 is

ω 7→ aω + b

cω + d. (3.14)

Note that while the action of the mapping class group SL(2,Z) on H2(T 2,Z) isfaithful, the action on the upper half plane H has kernel consisting of ±I. Byapplying a suitable element of SL(2,Z), we can arrange that ω = u + iv lie inthe fundamental domain of T1 for this action:

D = ω = u+ iv ∈ C : |u| ≤ 1/2, v > 0, |ω| ≥ 1.

The moduli space R1 is obtained from this fundamental domain by identify-ing edges. It can be shown that R1 has a Riemann surface structure and isholomorphically equivalent to the complex line C.

Thus we have two ways of looking at the Riemann surface structures on T 2,the Teichmuller space T1 when a choice of basis for H1(T 2; Z) is fixed, and themoduli space R1 when we ignore choice of basis. The discrete action of SL(2,Z)on T1 determines a branched covering T1 → R1.

If T 2 = C/Λ, then any map f : T 2 → M can be lifted to a doubly periodicmap f : C → M with periods in the lattice Λ. The standard coordinate z =x + iy on C descends to local coordinates on T 2, and the standard metricdx2+dy2 on T 2 descends to a flat metric on T 2 within the conformal equivalenceclass corresponding to ω. We can rescale the metric to (1/v)(dx2 + dy2) so thatit has total area one. In terms of the coordinates (t1, t2) on C defined by

x = t1 + ut2,

y = vt2,

the components of this flat metric and its inverse are

(ηab) =1v

(1 uu u2 + v2

), (ηab) =

1v

(u2 + v2 −u−u 1

),

while the area element is given by dA = dt1dt2. Hence if f ∈ C2(T 2,M),

|df |2 =∑a,b

ηab∂f

∂ti· ∂f∂tj

=1v

[(u2 + v2)

∣∣∣∣ ∂f∂t1∣∣∣∣2 − 2u

∂f

∂t1· ∂f∂t2

+∣∣∣∣ ∂f∂t2

∣∣∣∣2].

or equivalently,

|df |2 = v

∣∣∣∣ ∂f∂t1∣∣∣∣2 +

1v

∣∣∣∣ ∂f∂t2 − u ∂f∂t1∣∣∣∣2 ,

119

From this, it is apparent that the energy

E(f, ω) =12

∫ 1

0

∫ 1

0

[v

∣∣∣∣ ∂f∂t1∣∣∣∣2 +

1v

∣∣∣∣ ∂f∂t2 − u ∂f∂t1∣∣∣∣2]dt1dt2 (3.15)

is a smooth function on C2(T 2,M)× T1.In the following proof, it will be helpful to utilize the Euclidean coordinates

x1 = x√v

and x2 = y√v, in terms of which the energy of f is expressed as

E(f, ω) =12

∫∆

[∣∣∣∣ ∂f∂x1

∣∣∣∣2 +∣∣∣∣ ∂f∂x2

∣∣∣∣2]dx1dx2,

where ∆ is a fundamental domain for the lattice Λ.

Proposition 3.2.4. If (f, ω) is a critical point for

E : C2(T 2,M)× T1 −→ R,

then f is a weakly conformal harmonic map, and hence a parametrized minimalsurface.

Differentiation with respect to the first variable shows that if (f, ω) is a criticalpoint for E, f must be harmonic. So we need only consider the derivative withrespect to the second variable. If f : T 2 →M is any map, harmonic or not, wecan construct the quadratic differential

Ω(f, ω) =[∫

T 2

⟨∂f

∂z,∂f

∂z

⟩dx1dx2

]dz2,

which specializes to the Hopf differential of f when f is harmonic. This differ-ential is closely associated with a symmetric bilinear form

H(f, ω) : T0T2 × T0T

2 −→ R

that we define on the tangent space to the torus at an origin 0 by

H(f, ω) =∑

habdxadxb, where hab =

∫T 2

⟨∂f

∂xa,∂f

∂xb

⟩dx1dx2, (3.16)

which is easily seen to be positive definite. The trace of the quadratic form is2E(f), so the eigenvalues must be of the form

E(f, ω)± a, where 0 ≤ a ≤ E(f, ω),

and a short calculation shows that a = |Ω(f, ω)|.If we change the conformal structure from ω to ω, we must change the

Euclidean coordinates by a linear transformation defined by A ∈ SL(2,R),(x1

x2

)7→(x1

x2

)= A

(x1

x2

), and hence

(dx1

dx2

)= A

(dx1

dx2

).

120

It follows that the matrices H = (hij) and H = (hij) corresponding to H(f, ω)and H(f, ω) transform by to the rule,

H = AT HA, for A ∈ SL(2,R).

If A ∈ SO(2) this reduces to the transformation for a rotation of Euclideancoordinates for a fixed conformal structure.

Suppose now that H(f, ω) is degenerate; thus one of its eigenvalues is zero.We can rotate coordinates so that the x1 direction corresponds to the zeroeigenvalue. Then∫

T 2

⟨∂f

∂x1,∂f

∂x1

⟩dx1dx2 = 0 ⇒

⟨∂f

∂x1,∂f

∂x1

⟩≡ 0,

and the map f must degenerate to a curve. If f is harmonic with respect to ω,this curve must be a closed geodesic. The energy of the closed geodesic dependsupon its length and the choice of conformal structure ω. It is easily seen thatno such parametrized geodesic can be a critical point for E.

On the other hand, if H(f, ω) is nondegenerate, there exists an elementA ∈ SL(2,R) such that H = ATH0A, where H0 is a scalar multiple of theidentity matrix, and hence there is a conformal structure ω0 such that

H(f, ω) = E(f, ω0)〈·, ·〉.

Since H0 commutes with A, H = ATH0A = ATAH0. We can rotate theEuclidean coordinates chosen for ω so that ATA and H are diagonal, and then

ATA =(eλ 00 e−λ

), H = E(f, ω0)

(eλ 00 e−λ

),

for some choice of λ. Then E(f, ω) = (coshλ)E(f, ω0), and we see that the onlycritical points for E are not only harmonic, but also weakly conformal.

As a byproduct of the proof, we see that there are two types of harmonic tori.Those for which H(f, ω) is degenerate are simply parametrizations of smoothclosed geodesics, and these can never be conformal with respect to any conformalstructure. Those for which H(f, ω) is nondegenerate are either conformal ortheir energy can be decreased by replacing the conformal structure ω with anew conformal structure ω0.

Note that the differential dz descends from C to the torus C/Λ and henceholomorphic differentials on a torus must be of the form hdz2, where h is aholomorphic function on the torus. By the maximum modulus theorem, h mustbe constant. Hence the only possible Hopf differentials for harmonic tori arecdz2, where c is a complex constant. If the harmonic torus f : T 2 → M is notweakly conformal, its Hopf differential is never zero, so it cannot have branchpoints. However, it may have fold points as the following example shows.

Example 3.2.5. If f is an immersion, the fibers of the line bundles L and Lare linearly independent at every point. However, a harmonic map f can have

121

points p at which the fibers L and L coincide. At such points the rank of f canbe at most one.

Indeed, there is a degree zero harmonic map f : T 2 → S2 when S2 is giventhe standard Riemannian metric of constant curvature one, which has “foldpoints” along two circles parallel to the equator. To see how this is constructed,we first note that the metric on S2 ⊂ R2 with equation x2 + y2 + z2 = 1is expressed in spherical coordinates z = cosφ, θ being the standard angularcoordinate in the (x, y)-plane, is

ds2 = (cos2 φ)dθ2 + dφ2 = sech2u(dθ2 + du2),

where u and φ are related by the equation tanh(u/2) = tan(φ/2). In terms ofthe standard coordinates (t1, t2) on T 2 which correspond to the factorizationT 2 = S1 × S1, the mapping f : T 2 → S2 can expressed as

θ(t1, t2) = t2, φ(t1, t2) = φ(t1),

where φ is a (nonconstant speed) parametrization of the geodesic θ = (constant).Note that the circle φ = (constant) has curvature κ = 1/ cosφ and normal

curvature κn = 1. From the equation κ2g + κ2

n = κ2, where κg is the geodesiccurvature, implies that κg = ± tanφ. Moreover, the curve is traversed withconstant speed cosφ. Hence

0 =D

∂t1

(∂f

∂t1

)+

D

∂t2

(∂f

∂t2

)=d2φ

dt2+ (tanφ)(cos2 φ) =

d2φ

dt2+

12

sin(2φ).

The differential equation we must solve to obtain a harmonic map (the pendulumequation except for constant factors) is equivalent to the first order system

dφ

dt= ψ,

dψ

dt= −1

2sin(2φ).

Eliminating dt yields

dψ

dφ= − (1/2) sin(2φ)

ψwhich integrates to ψ2 − 1

2cos(2φ) = (constant).

Various choices of constant yield harmonic maps for appropriate conformalstructures on T 2.

Note that the antipodal map A : S2 → S2 induces an orientation reversingmap A : T 2 → T 2 such that f A = A f . We can take the quotient in bothdomain and range, obtaining thereby a harmonic map from a Klein bottle intothe real projective plane RP 2, which has as its image a Mobius band.

3.3 Minimal surfaces of higher genus

Finally, we consider the case where Σ is a sphere with g handles and g ≥ 2.Classifying the conformal structures on Σ is now somewhat more challenging.

122

We let Metk(Σg) denote the space of L2k Riemannian metrics

η =2∑

a,b=1

ηabduadub

on Σ, an open subset of a Hilbert space, and let Metk0(Σg) denote the submani-fold of metrics of constant Gaussian curvature and total area one. It is not diffi-cult to conclude from the Uniformization Theorem that given any η ∈ Metk(Σg),there is a rescaling λ2η, where λ2 > 0, which lies in Metk0(Σg). Moreover, onecan show that this rescaling is unique and we thus obtain a strong deformationretraction

π : Metk(Σg) −→ Metk0(Σg). (3.17)

In addition, we consider

Diffk+1+ (Σg) = φ ∈ L2

k+1(Σ,Σ) such thatφ is an orientation-preserving diffeomorphism ,

which is a group under composition, and its subgroup

Diffk+10 (Σg) = φ ∈ Diffk+1

+ (Σg) : φ is homotopic to the identity .

Unfortunately, although Diffk+1+ (Σg) and Diffk+1

0 (Σg) are smooth Hilbert man-ifolds, the group multiplications are not smooth, so these are not infinite-dimensional Lie groups. However, the maps

ψ : Diffk+1+ (Σg)×Metk(Σg) −→ Metk(Σg), ψ(φ, η) = φ∗η,

and its restriction

ψ : Diffk+1+ (Σg)×Metk0(Σg) −→ Metk0(Σg),

are indeed smooth. Moreover, ψ(φ, π(η)) = πψ(φ, η), where π is the projection(3.17) into constant curvature metrics. Of course, either of these maps can befurther restricted to the subgroup Diffk+1

0 (Σg).

Lemma 3.3.1. The continuous action of Diffk+10 (Σg) on Metk0(Σg) is free.

Moreover, the image of the map

ψ : Diffk+1+ (Σg)×Metk0(Σg) −→ Metk0(Σg)×Metk0(Σg), ψ(φ, η) = (η, ψ(φ, η)),

(3.18)is closed.

Indeed, if the action of Diffk+10 (Σg) on Metk0(Σg) were not free, there would be

two distinct isometries

id, φ : (Σg, φ∗η) −→ (Σg, η),

123

both homotopic to the identity. Each would be harmonic, contradicting Hart-man’s Theorem 3.1.2 which asserts that the harmonic map between two compactsurfaces of negative curvature in a given homotopy class is unique. We leavethe second statement as an easy exercise for the reader.

The orbit spaces,

Tg =Metk0(Σg)

Diffk+10 (Σg)

and Rg =Metk0(Σg)

Diffk+1+ (Σg)

,

are called the Teichmuller space and the Riemann moduli space of conformalstructures on Σg, respectively. The fact that the image of the map ψ in (3.18)is closed is equivalent to the orbit space Tg being Haussdorf.

Theorem 3.3.2. If g ≥ 2, the Teichmuller space Tg is a manifold of dimension6g − 6.

This is essentially due to Earle and Eells [15], and a modern argument usingBanach manifolds is presented in Fischer and Tromba [20]. Our argument willmostly follow Chapter 2 of [73]. Our goal is to construct local coordinates in Tgabout a base metric η ∈ Metk0(Σg). Let G = Diffk+1

0 (Σg). According to a well-known result regarding group actions (namely Theorem 2.9.10 in Varadarajan[81]), it suffices to construct a submanifold S ⊂ Metk0(Σg) such that η ∈ S and

1. TηMetk0(Σg) ∼= TηS ⊕ Tη(G · η) and

2. any G-orbit intersects S in only one point.

Such a submanifold is called a slice for the action, and makes it possible toconstruct an open neighborhood U of η in S and a diffeomorphism from U ×Gto an open neighborhood of the orbit through η.

To construct this slice, we first note that if

X =2∑a=1

Xa∂

∂ua

is a smooth vector field on Σ with local one-parameter subgroup φt : t ∈ R,it follows from a familiar calculation that

d

dt(φ∗t η)

∣∣∣∣t=0

= LXη = · · · =2∑

a,b=1

Xa;bduadub,

where the Xa;b’s are the components of the covariant derivative of X with re-spect to the metric η. We let 〈·, ·〉 denote the G-invariant L2 inner product onTηMetk(Σg) such that

〈η, η〉 =∫

Σ

1λ2

2∑a,b=1

ηabηabdu1du2 and let ‖η‖ =

√〈η, η〉.

124

We suppose that we have chosen coordinates so that the components of our basemetric η are ηab = δabλ

2, for some positive function λ2, and consider when aone-parameter family of metrics,

ηab(t) = ηab + tηab for t ∈ (ε, ε),

is perpendicular with respect to 〈·, ·〉 at t = 0 to the orbit through η. Anintegration by parts shows that

〈LXη, η〉 =∫

Σ

1λ2

2∑a,b=1

ηabXa;bdu1du2 = −

∫Σ

1λ2

2∑a,b=1

ηab;bXadu1du2,

so that η is perpendicular to the G-orbit when∑ηab;b = 0. This suggests a

slice for the action of G on the larger space Metk(Σg), namely

Sη =

η + η : η ∈ TηMetk(Σg), ‖η‖ < ε and

2∑b=1

ηab;b = 0

.

To obtain a section for the action on the smaller space Metk0(Σg), we utilize thefollowing lemma:

Lemma 3.3.3. If

ηab(t) = ηab + tηab for t ∈ (ε, ε)

is a one-parameter family of metrics of constant Gaussian curvature K and totalarea one, ηab;b = 0 and

∆η(0)(Tr(η)) = −2KTr(η), where Tr(η) =1λ2

(η11 + η22)

and ∆η(0) is the Laplace operator for the base metric η(0).

To prove this, we can use geodesic normal coordinates centered for η at a givenpoint p ∈ Σ. In terms of such coordinates, the curvature tensor of η(t) is givenby

K det(ηab(t)) = R1212(t) =12

(2η12,12(t)− η11,22(t)− η22,11(t))

+ (higher order terms),

where the commas denote coordinate derivatives. Differentiating with respectto t and setting t = 0, we obtain

Kλ2(η11 + η22) =12

(2η12;12 − η11;22 − η22;11) ,

where we have replaced ordinary derivatives by covariant derivatives and haveevaluated at p. Finally, we use the identity

∑ηab;b = 0 to simplify the right-

hand side,

Kλ2(η11 + η22) = −12

2∑a,b=1

ηaa;bb

,

125

which yields the statement of the lemma:

1λ2

2∑a,b=1

(1λ2ηaa

);bb

= −2K1λ2

(η11 + η22)

Since K is negative, it follows from the Lemma and the maximum principleapplied to the operator ∆η(0) + 2K that η11 + η22 = 0. This, together with theidentity

∑ηab;b = 0 implies that if w = u1 + iu2,

∂

∂w(η11 + iη12) = 0,

and hence(η11 + iη12)dw2

is a holomorphic quadratic differential. Thus we set

Sη = η + η : η ∈ TηMetk(Σg), ‖η‖ < ε and(η11 + iη12)dw2 is a holomorphic quadratic differential ,

a convex open subset of an affine subspace of a Hilbert space. One of theimplications of the Riemann-Roch Theorem from Riemann surface theory isthat the space of holomorphic quadratic differentials on a Riemann surface ofgenus g has complex dimension 3g − 3 or real dimension 6g − 6, so the affinespace in which Sη lies has dimension 6g − 6.

A straightforward (if slightly tedious) calculation shows that the identitymap

id : (Σg, η) −→ (Σg, η + η)

is harmonic with Hopf differential (1/2)(η11+iη12)dw2. If theG-orbit intersectedSη in more than one point, we would have homotopic harmonic maps from(Σg, η) to the same target with two different Hopf differentials, contradictingTheorem 3.1.2 on uniqueness for harmonic maps. The section Sη is only tangentto the submanifold Metk0(Σg), but since the projection π to the space Metk0(Σg)commutes with the G-action, S = π(Sη) is a slice for the G-action on Metk0(Σg)of dimension 6g − 6, completing the proof of the theorem.

Remark 3.3.4. In contrast to the Teichmuller space Tg, the Riemann modulispace Rg is not even a manifold in general, but an orbifold. It is the quotientof Teichmuller space by the properly discontinuous action of the mapping classgroup Γg = Diff+(Σg)/Diff0(Σg).

Choose a base Riemann surface (Σ, η0) of genus g and let O(κ2) denote thespace of holomorphic quadratic differentials on (Σ, η0), a real vector space ofdimension 6g − 6. We can define a map

Ψ : Tg → O(κ2) by Ψ(Σ, η) = Ω(f),

126

where Ω(f) is the Hopf differential of the unique harmonic map f from (Σ, η0)to (Σ, η).

Theorem 3.3.5. (Teichmuller) If g ≥ 2, the map Ψ : Tg → O(κ2) is adiffeomorphism, and hence Tg is diffeomorphic to R6g−6.

Teichmuller’s original proof that Tg is homeomorphic to R6g−6 was based uponthe theory of quasiconformal maps, and was later simplified by Bers, as pre-sented in [37]. Later a proof was found using harmonic maps, and we willreturn to discuss it later.

Let η0 ∈ Metk0(Σg) be a base Riemannian metric. If η is any element of Tg, itfollows from Theorems 3.1.1 and 3.1.2 that there is a unique harmonic map

s(η) : (Σ, η)→ (Σ, η0),

which depends smoothly on η in terms of the above coordinates. Using a variantof the Bochner Lemma to be treated in the next section, Schoen and Yau wereable to prove that s(η) is a diffeomorphism ([73], Chapter 1, §8). Thus we candefine a map

Metk0(Σg)→ Metk0(Σg) by σ(η) = [s(η)−1]∗(η).

Since φ s(φ∗η) = s(η), the metric s(η) defines a section σ : Tg → Metk0(Σg).We can now consider the energy as a map of two variables E : C2(Σ,M)×

Metk0(Σg)→ R defined by

E(f, (ηab)) =12

∫Σ

∑a,b

ηab⟨∂f

∂ua,∂f

∂ub

⟩√ηdu1du2, (3.19)

where (ηab) is the matrix inverse of (ηab) and η = det(ηab), the integrand beingindependent of choice of local coordinates. If φ ∈ Diffk+1

0 (Σg), then

E(f φ, φ∗η) = E(f, η),

and hence E descends to a map on the space of orbits

C2(Σ,M)×Metk0(Σg)Diffk+1

0 (Σg).

However, using the map σ described above, one can exhibit this quotient assimply a product C2(Σ,M)× Tg.

Theorem 3.3.6. If (f, ω) is a critical point for the map

E : C2(Σ,M)× Tg −→ R,

then f is harmonic and weakly conformal with respect to ω, and hence aparametrized minimal surface.

127

Differentiation with respect to f shows that f is harmonic. We need to calculatethe derivative with respect to the metric, so we take a perturbation of a givenmetric with support in a given isothermal coordinate system (u1, u2) on Σ.Suppose that the perturbation is given by the formula

ηab(t) = ηab + tηab = ηab + tψ(u1, u2)ρab(u1, u2),

where ψ(x1, x2) has compact support, the variation in the metric is trace-free(η11 + η22 = 0) and the initial metric is given by the formulae

ηab = δabλ2,

√η = λ2,

for some conformal factor λ2. Then a direct calculation shows that

d

dtηab(t) = ηab,

d

dtη(t)

∣∣∣∣t=0

= λ2(η11 + η22) = 0,

From this, we can easily calculate

d

dt

(√ηη11 √

ηη12

√ηη21 √

ηη22

)∣∣∣∣t=0

= λ−2

(η22 −η12

−η21 η11

).

Thus we find that

d

dtE(f, ηab(t))

∣∣∣∣t=0

=∫

Σ

∑a,b

d

dt

√ηηab

∣∣∣∣t=0

⟨∂f

∂ua,∂f

∂ub

⟩du1du2

= −∫

Σ

[η11

λ2

(⟨∂f

∂u1,∂f

∂u1

⟩−⟨∂f

∂u2,∂f

∂u2

⟩)+

2η12

λ2

⟨∂f

∂u1,∂f

∂u2

⟩]du1du2

= −4∫

Σ

Re[(η11 + iη12)

⟨∂f

∂w,∂f

∂w

⟩]1λ2du1du2, (3.20)

where w = u1 + iu2. Since this is true for all trace-free variations in the metric,we conclude that⟨

∂f

∂u1,∂f

∂u1

⟩−⟨∂f

∂u2,∂f

∂u2

⟩= 0 =

⟨∂f

∂u1,∂f

∂u2

⟩,

and hence the Hopf differential Ω(f) must vanish. This finishes the proof of thetheorem.

Of course, it is also true that E(f φ, φ∗η) = E(f, η) for φ in the larger groupof orientation-preserving diffeomorphisms Diffk+1

+ (Σg), but this group does notact freely on C2(Σ,M)×Metk0(Σg) and the quotient space

M(Σ,M) =C2(Σ,M)×Metk0(Σg)

Diffk+1+ (Σg)

is not a manifold, but only an orbifold in general. We have two ways of lookingat parametrized minimal surfaces of genus g, as critical points for the energy

E :M(Σ,M) −→ R

128

or as Γg-orbits of critical points for

E : C2(Σ,M)× Tg −→ R,

where Γg = Diffk+1+ (Σg)/Diffk+1

0 (Σg) is the mapping class group.

3.4 The Bochner Lemma

The key step behind the existence theory for harmonic is the Bochner Lemmawhich gives an estimate for the Laplacian of the energy density in terms of cur-vature. We describe the Bochner Lemma in this section, following the treatmentin [45]. Let Σ and M be compact Riemannian manifolds with metrics (ηab) and(gij) respectively, with M as usual being isometrically imbedded in Euclideanspace RN . We suppose that

f : Σ −→M ⊆ RN

is a smooth map and for given choice of coordinates (u1, . . . , um) on Σ, weconsider the vector-valued maps

fa =∂f

∂ua∈ RN which have components f ia =

∂f i

∂ua,

with respect to coordinates (x1, . . . , xn) on M . Recall that the energy densityis given by

e(f) =12

∑ηabfa · fb =

12

∑gijη

abf iafjb .

Finally, we letRΣ = (Rabcd) and RM = (Rijkl)

be the curvature tensors of Σ and M , and let RicΣ = (Rab) be the Ricci cur-vature of Σ. Recall that if we divide the Laplacian of the vector-valued mapf : Σ→ RN into tangential and normal components,

∆ηf = (∆ηf)> + (∆ηf)⊥,

the condition that f be harmonic is just (∆ηf)> = 0. On the other hand,the normal component of the Laplacian is expressed in terms of the secondfundamental form

α(f)(p) : TpM × TpM −→ NpM

of M within RN by

(∆ηf)⊥(p) =∑

ηabα(f)(p)(fa, fb).

Theorem 3.4.1 (Bochner Lemma). If f : Σ→M is a smooth map, then

∆η(e(f)) = |∇df |2 + 〈d[(∆ηf)>], df〉

+∑

ηacηbdRabfc · fd −∑

ηacηbdRijklfiaf

jb f

kc f

ld. (3.21)

129

where ∇ denotes covariant derivative with respect to the connection in T ∗Σ ⊗f∗TM . Hence if f is harmonic,

∆η(e(f)) = |∇df |2 +∑

ηacηbdRabfc · fd −∑

ηacηbdRijklfiaf

jb f

kc f

ld. (3.22)

To prove this, we use normal coordinates centered a point p ∈ Σ and similarcoordinates at f(p) in M . Then

∆η(e(f)) =∑|fa;b|2 +

∑〈fa, fa;bb〉

=∑|fa;b|2 +

∑〈fa, fb;ba〉 −

∑〈fa, Rbcabfc〉

=∑|fa;b|2 +

∑〈fa, (∆ηf)a〉+

∑Rab〈fa, fb〉. (3.23)

On the other hand,

〈fa, (∆η(f))a〉 = 〈fa, [(∆ηf)>]a〉+ 〈fa, [(∆ηf)⊥]a〉

= 〈fa, [(∆ηf)>]a〉+∑〈fa, (α(f)(fb, fb)a〉.

Now we use the fact that

〈fa, α(f)(f ib , fib)〉 = 0

to conclude that

〈fa, (∆η(f))a〉 = 〈fa, [(∆ηf)>]a〉 −∑〈∆η(f), (α(f)(fb, fb)〉

= 〈fa, [(∆ηf)>]a〉 −∑〈(α(f)(fa, fa), (α(f)(fb, fb)〉. (3.24)

Finally, we note that∑|fa;b|2 = |∇df |2 +

∑〈α(f)(fa, fb), α(f)(fa, fb)〉. (3.25)

Finally, we substitute (3.24) and (3.25) into (3.24) and obtain

∆η(e(f)) = |∇df |2 + 〈d[(∆ηf)>], df〉+∑

Rabfa · fb

−[∑〈α(f)(fa, fa), (α(f)(fb, fb)〉 −

∑〈α(f)(fa, fb), α(f)(fa, fb)〉

].

This last equation implies (3.21) by the Gauss equation.

If Σ is a compact surface with Gaussian curvature K, then (3.22) simplifies to

∆η(e(f)) = |∇df |2 + K|df |2 −∑

Rijklfiaf

jb f

ka f

lb. (3.26)

In terms of isothermal coordinates (u, v) = (u1, u2) on Σ, we can write∑ηabdu

adub = λ2(du2 + dv2), a(f) =1λ2

∣∣∣∣∂f∂u ∧ ∂f∂v∣∣∣∣ .

130

Then (3.26) becomes

∆η(e(f)) ≥ |∇df |2 + 2Ke(f)− 2K(σ)a(f)2, (3.27)

where K(σ) is the sectional curvature of the two-plane generated by ∂f/∂u and∂f/∂v. It follows from inequality (3.10) that if K0 ≥ 0, then

K(σ) ≤ K0 ⇒ ∆η(e(f)) ≥ |∇df |2 + 2Ke(f)− 2K0e(f)2, (3.28)

equality holding when f is conformal.

Corollary 3.4.2. If M has nonpositive sectional curvatures, there are no har-monic maps from the two-sphere S2 into M and any harmonic map from thetorus T 2 into M must be totally geodesic. If M has negative sectional curva-tures, there are no harmonic maps from S2 or T 2 into M .

Proof: If Σ = S2, we can choose the Riemannian metric on Σ so that K ≡ 1.The continuous function e(f) must assume a maximum at some point. At thispoint the Bochner inequality shows that ∆(e(f)) > 0, a contradiction.

If Σ = T 2, we can choose the Riemannian metric on Σ so that K ≡ 0. Inthis case, the inequality must be an equality. Hence ∇df = 0 which implies that

∂f

∂u1and

∂f

∂u2

are parallel sections of the pullback of the bundle f∗TM . This implies of coursethat f is totally geodesic.

The case where M has negative sectional curvatures is proven in a similarfashion.

3.5 The α-energy

In order to apply critical point theory to a function on an infinite-dimensionalmanifold, we need the function to be continuous and satisfy condition C ofPalais and Smale. For the usual energy, however, the latter condition wouldrequire that we complete the space C2(Σ,M) with respect to the L2

1-topology,a topology which is too weak for the usual techniques of global analysis, sincethe Sobolev inequality barely fails to show that L2

1 functions are continuous.Thus Sacks and Uhlenbeck [68], [69] were led to consider perturbations of

the energy. Among the simplest such perturbations are functions

F : C∞(Σ,M)→ R defined by F (f) =∫

Σ

φ(|df |2)dA,

where φ : R→ R is a smooth function. By calculating the differential of F andintegrating by parts (assuming that f is sufficiently differentiable), we obtainthe Euler-Lagrange equation for critical points

D

∂u

(φ′(|df |2)

∂f

∂u

)+D

∂v

(φ′(|df |2)

∂f

∂v

)= 0, (3.29)

131

whenever (u, v) is a set of isothermal coordinates on Σ.

Example 3.5.1. We could take φ(t) = tα, where α > 1, which has continuousderivatives up to order two when regarded as a function on L2α

1 (Σ,M), a Banachspace which does lie within Sobolev range. Thus for a given Riemannian metricη on Σ, we could define

Eα,η : L2α1 (Σ,M)→ R by Eα,η(f) =

12

∫Σ

|df |2αdA, (3.30)

However, the critical points of Eα,η are not necessarily smooth. Indeed, in thespecial case where M = R, the Euler-Lagrange equation

∂

∂u

(|df |2α−2 ∂f

∂u

)+

∂

∂v

(|df |2α−2 ∂f

∂v

)= 0

can be rewritten in terms of polar coordinates (r, θ) as

1r

∂

∂r

(r|df |2α−2 ∂f

∂r

)+

1r2

∂

∂θ

(|df |2α−2 ∂f

∂θ

)= 0,

which is satisfied by f(r, θ) = r(2α−2)/(2α−1). Although this function is not C1,it can be checked that it does lie in L2α

1 . Thus we see that critical points in L2α1

for this choice of φ are not necessarily C∞.

However, the slight modification in which φ(t) = (1 + t)α provides a functionwhich has C∞ critical points, as we will see later, and that is the choice thatwe will make.

Definition. Suppose that M is a smooth manifold with Riemannian metricg. Given a Riemannian metric η on Σ, the α-energy is the function Eα,η :L2α

1 (Σ,M)→ R defined by

Eα,η(f) =12

∫Σ

(1 + |df |2)αdA, (3.31)

for α > 1, where dA is calculated with respect to η and |df | is calculated withrespect to η and g. For ω ∈ T , the (α, ω)-energy Eα,ω is just the function Eα,η,where η is chosen to be the unique constant curvature metric of total area oneon Σ within the conformal class ω.

By the normalization we have chosen for the metric, the α-energy Eα,ω ap-proaches Eω + (1/2) as α→ 1.

Note that the (α, ω)-energy on M is a composition of the map ωi, induced viaTheorem 1.4.7 (the so-called ω-lemma) by an isometric imbedding i : M → RN ,with the (α, ω)-energy on RN . The latter map, Eα,ω : L2α

1 (Σ,RN ) → R, isclearly continuous, and it is also C2 with derivatives

dEα,ω(f)(V ) = α

∫Σ

(1 + |df |2)α−1df · dV dA (3.32)

132

and

d2Eα,ω(f)(V1, V2) = α

∫Σ

(1 + |df |2)α−1dV1 · dV2dA

+ 2α(α− 1)∫

Σ

(1 + |df |2)α−2(df · dV1)(df · dV2)dA, (3.33)

by the same argument we used in Example 1.2.3. Since the composition of C2

maps is C2 we see that the map Eα,ω on L2α1 (Σ,M) is also C2.

We will see shortly that Eα,ω satisfies condition C and its critical points aresmooth. If we let ω ∈ T vary and set Eα(f, ω) = Eα,ω(f), we get a C2 function

Eα : L2α1 (Σ,M)× Tg → R.

Definition. A critical point f ∈ L2α1 (Σ,M) for Eα,ω is called an (α, ω)-

harmonic map, or sometimes an α-harmonic map. If (f, ω) ∈ L2α1 (Σ,M)×T is

a critical point for Eα, f is called a (parametrized) α-minimal surface.

In complete analogy with ω-harmonic maps, we can calculate the first variationformula for (α, ω)-harmonic maps. Indeed, suppose that Σ is a compact Rie-mann surface. By the same argument that led to (3.29), we obtain the formula

dEα,ω(f)(V ) = −∫

Σ

[⟨D

∂u

(γ2 ∂f

∂u

), V

⟩+⟨D

∂v

(γ2 ∂f

∂v

), V

⟩]dudv,

where γ2 = (1+|df |2)α−1. In the integrand, (u, v) can be any choice of conformalcoordinates. This leads to the Euler-Lagrange equation

D

∂u

(γ2 ∂f

∂u

)+D

∂v

(γ2 ∂f

∂v

)= 0. (3.34)

This equation can also be written in terms of the standard Laplace operatoracting on f ,

D

∂u

(∂f

∂u

)+D

∂v

(∂f

∂v

)= −(α− 1)

[∂

∂u(log(1 + |df |2))

∂f

∂u+

∂

∂v(log(1 + |df |2))

∂f

∂v

]. (3.35)

Our next goal is to show that the function Eα,ω : L2α1 (Σ,M) → R satisfies

condition C. Recall that condition C asserts that if fi is a sequence of pointsin L2α

1 (Σ,M) such that

Eα,ω(fi) ≤ E0 where E0 is some constant, and ‖dEα,ω(fi)‖ → 0,

then fi possesses a subsequence which converges in L2α1 (Σ,M) to a critical

point for Eα,ω.

133

Theorem 3.5.2. If α > 1, the function Eα,ω : L2α1 (Σ,M) → R satisfies

condition C.

The proof is very similar to the one given before for the action integral in thetheory of smooth closed geodesics. If V is tangent to L2α

1 (Σ,M),

dEα,ω(f)(V ) = 2α∫

Σ

(1 + |df |2)α−1

[⟨DV

∂u,∂f

∂u

⟩+⟨DV

∂v,∂f

∂v

⟩]dudv,

or equivalently,

dEα,ω(f)(V ) = −2α∫

Σ

⟨D

∂u

(γ2 ∂f

∂u

)+D

∂v

(γ2 ∂f

∂v

), V

⟩dudv,

for V ∈ Tf (L2α1 (Σ,M)), where γ2 = (1 + |df |2)α−1 and D denotes covariant

derivative. More generally, we can consider an element V ∈ Tf (L2α1 (Σ,RN )) and

project it into the tangent space to L2α1 (Σ,M). Starting with the orthogonal

projection P : i∗TRN → TM into the tangent space, we use the ω-Lemma toconstruct a smooth map

ωP : L2α1 (Σ, i∗TRN ) −→ L2α

1 (Σ, TM).

Lemma 3.5.3. If V ∈ Tf (L2α1 (Σ, i∗TRN ),

dEα,ω(f)(ωP (V )) = α

∫Σ

(1 + |df |2)α−1

[∂V

∂u· ∂f∂u

+∂V

∂v· ∂f∂v

−V ·(α

(∂f

∂u,∂f

∂u

)+ α

(∂f

∂v,∂f

∂v

))]dudv,

where α : TM × TM → NM is the second fundamental form of M in RN .

In the proof, we can suppose that f and V are C∞, since the C∞ maps are densein L2α

1 . We write V = ωP (V ) + V ⊥, where ωP (V ) and V ⊥ are the componentsof V which are tangent and normal to M . Then(

∂

∂u

(γ2 ∂f

∂u

))· V =

(∂

∂u

(γ2 ∂f

∂u

))· ωP (V ) +

(∂

∂u

(γ2 ∂f

∂u

))· V ⊥

=(D

∂x

(γ2 ∂f

∂x

))· ωP (V ) + γ2α

(∂f

∂x,∂f

∂x

)· V ⊥,

and similarly(∂

∂v

(γ2 ∂f

∂v

))· V =

(D

∂v

(γ2 ∂f

∂v

))· ωP (V ) + γ2α

(∂f

∂v,∂f

∂v

)· V ⊥,

where the dot denotes the dot product in the ambient Euclidean space RN . It

134

follows that

dEα,ω(f)(ωP (V ))

= −2α∫

Σ

[(∂

∂u

(γ2 ∂f

∂u

)+

∂

∂v

(γ2 ∂f

∂v

))· V

+γ2

(α

(∂f

∂u,∂f

∂u

)+ α

(∂f

∂v,∂f

∂v

))· V ⊥

]dudu.

An integration by parts now yields the statement of the Lemma.

Lemma 3.5.4. There is a constant C depending only on M such that

‖ωP (V )‖L2α1≤ C(1 + Eα,ω(f))‖V ‖L2α

1for V ∈ Tf (L2α

1 (Σ, i∗TRN ).

This is a straightforward consequence of applying the Leibniz rule to the equa-tion

(ωP (V ))(p) = Pf(p)V (p),

where Pf(p) is the projection from RN into the tangent space Tf(p)M , to obtain

∂

∂u(ωP (V ))(p) =

(∂P f∂u

)(V (p)) + Pf(p)

(∂V

∂u(p)),

∂

∂v(ωP (V ))(p) =

(P ∂f∂v

)(V (p)) + Pf(p)

(∂V

∂v(p)).

Thus that after possibly replacing the usual L2α1 norm with an equivalent norm,

we can write‖ωP (V )‖L2α

1= ‖ωP (V )‖L2α + ‖dωP (V )‖L2α ,

but

‖dωP (V )‖L2α =∥∥∥∥(∂P f∂u

)(V (p))du+ Pf(p)

(∂V

∂u(p))du

∥∥∥∥L2α

+∥∥∥∥(∂P f∂v

)(V (p))dv + Pf(p)

(∂V

∂v(p))dv

∥∥∥∥L2α

≤ C1 sup‖(V (p))‖ : p ∈ Σ ‖df‖L2α + C2 ‖DV ‖L2α ,

where C1 and C2 are positive constants. Thus we see that

‖ωP (V )‖L2α1≤ C3

[‖V ‖L2α

1Eα,ω(f) + C4 ‖DV ‖L2α + C5‖V ‖L2α

],

where C3, C4 and C5 are yet other positive constants and we have used thefact that the C0-norm is less than some constant times the L2α

1 -norm. The lastestimate yields the statement of the lemma.

Let us turn now to the proof of Theorem 3.5.2 itself. Let fi be a sequence fromL2α

1 (Σ,M) ⊂ L2α1 (Σ,RN ) such that Eα,ω(fi) is bounded and ‖dEα,ω(fi)‖ →

135

0. By the Sobolev Lemma 1.4.4 each fi ∈ C0(Σ,M). Moreover, since M iscompact, fi is uniformly bounded, and it follows from the Holder estimate inLemma 1.4.4 that fi is equicontinuous. Therefore, the Arzela-Ascoli Theoremfrom real analysis implies that a subsequence of fi converges uniformly to acontinuous map f∞ : Σ→M .

Since Eα,ω(fi) is bounded and fi is bounded in C0, it follows that fi isbounded in L2α

1 (Σ,RN ). It then follows from Lemma 3.5.3 that ωP (fi − fj) isbounded in L2α

1 (Σ,RN ). Since ‖dEα,ω(fi)‖ → 0,

|dEα,ω(fi)(ωP (fi − fj))− dEα,ω(fj)(ωP (fi − fj))| → 0 as i, j →∞.

Using Lemma 3.5.2, we can rewrite this as∣∣∣∣2α ∫Σ

(1 + |dfi|2)α−1

[∂fi∂u·(∂fi∂u− ∂fj∂u

)+∂fi∂v·(∂fi∂v− ∂fj

∂v

)

−(α

(∂fi∂u

,∂fi∂u

)+ α

(∂fi∂v

,∂fi∂v

))· (fi − fj)

]dudv

−2α∫

Σ

(1 + |dfj |2)α−1

[∂fj∂u·(∂fi∂u− ∂fj∂u

)+∂fj∂v·(∂fi∂v− ∂fj

∂v

)−(α

(∂fj∂u

,∂fj∂u

)+ α

(∂fj∂v

,∂fj∂v

))· (fi − fj)

]dudv

∣∣∣∣→ 0

as i and j approach infinity. Note that since energy is bounded,∣∣∣∣∫Σ

α

(∂fi∂u

,∂fi∂u

)· (fi − fj)dudv

∣∣∣∣ ≤ (constant) sup |fi − fj | → 0

and∣∣∣∣∫

Σ

α

(∂fi∂v

,∂fi∂v

)· (fi − fj)dudv

∣∣∣∣ ≤ (constant) sup |fi − fj | → 0

as i and j approach infinity, and hence∣∣∣∣∫Σ

(1 + |dfi|2)α−1〈dfi, dfi − dfj〉dA

−∫

Σ

(1 + |dfj |2)α−1〈dfj , dfi − dfj〉dA∣∣∣∣→ 0

as i and j approach infinity.To proceed further, we need:

Lemma 3.5.5. If α > 1, there is a constant c ≥ (1/16) such that

(|v|2α−2v − |w|2α−2w) · (v − w) ≥ c|v − w|2α, for all v, w ∈ R2N+1.

136

Proof: It suffices to establish this inequality when both v and w are nonzero.If we set f(v) = |v|2α, then f is C2 on R2N+1 − 0, and a direct calculationshows that

Df(v)(w) = 2α|v|2α−2v · w,D2f(v)(w,w) = 4α(α− 1)|v|2α−4(v · w)2 + 2α|v|2α−2w · w ≥ 2α|v|2α−2w · w.

Hence

Df(v)(v − w)−Df(w)(v − w) =∫ 1

0

D2f(w + t(v − w))(v − w, v − w)dt

≥ 2α∫ 1

0

|w + t(v − w)|2α−2|v − w|2dt ≥ c|v − w|2α,

for some constant c > 0. To see that c can be chosen to be larger than 1/16,note that by the triangle inequality, either

|w| ≥ 12|v − w| ⇒ |w + t(v − w)| ≥ 1

2|w| ≥ 1

4|v − w| for t ∈ [0, 1/4],

or

|v| ≥ 12|v − w| ⇒ |v + (1− t)(w − v)| ≥ 1

2|v| ≥ 1

4|v − w| for t ∈ [3/4, 1].

This proves the lemma.

To apply Lemma 3.5.5, we set

v =(

1,∂fi∂u

,∂fi∂v

), w =

(1,∂fj∂u

,∂fj∂v

).

Then Lemma 3.5.5 implies that∫Σ

|dfi − dfj |2αdA→ 0 as i, j →∞.

Since fi → f∞ in C0, it follows that fi is a Cauchy sequence in L2α1 (Σ,RN ).

By completeness of the Banach space L2α1 (Σ,RN ), there exists an element

f∞ ∈ L2α1 (Σ,RN ) such that fi → f∞ in L2α

1 (Σ,RN ). Clearly, f∞ = f∞ ∈L2α

1 (Σ,M), and Theorem 3.5.2 is proven.

Of course, there are versions of Theorem 3.5.2 which hold for perturbations ofthe α-energy. For example, if α > 1 and ψ ∈ L2

k(Σ,RN ) for k ≥ 2, the perturbedfunction

Eα,ψ,ω : L2α1 (Σ,M)→ R, defined by Eα,ψ,ω(f) = Eα,ω(f) +

∫Σ

f · ψdA

satisfies condition C. In fact, even more generally, we could consider the functionEβα,ψ,ω : L2α

1 (Σ,M)→ R defined by

Eβα,ψ,ω(f) =12

∫Σ

(β2 + |df |2)αdA+∫

Σ

f · ψdA. (3.36)

137

Theorem 3.5.5. If α > 1 and β ∈ [0, 1], then the function Eβα,ψ,ω defined by(3.36) satisfies condition C.

The proof is a straightforward modification of that given for Theorem 3.5.2.Note that the argument works even if β = 0, in which case the critical pointsare not necessarily smooth.

Thus we can use Corollary 1.11.3 to show that any component of L2α1 (Σ,M)

contains an element f which minimizes Eα,ω or more generally Eβα,ψ,ω. More-over, we can use Theorem 1.11.2 to prove existence of minimax critical pointsfor these functions corresponding to various algebraic topology constraints. Wenote that

E(f) ≤ E0α(f) ≤ Eβα(f)− β2α

2, (3.37)

the second of these inequalities following from the fact that

ψ(t) = (β2 + t2)α − (β2α + t2α), then ψ′(t) > 0, for t > 0,

an immediate consequence of differentiating ψ with respect to t.Of course, our goal is to use these existence results to obtain existence of

ω-harmonic maps in the limit as the various perturbations are turned off. Thus,for example, we could let β → 0 first, obtaining a function that has simplerbehavior under rescaling, and then let α → 1. Condition C is only lost whentaking the second limit.

3.6 Regularity of (α, ω)-harmonic maps

At some point, it becomes useful to know that the (α, ω)-harmonic maps con-structed by means of Condition C are actually smooth maps. Indeed, if theambient manifold M is C∞, so is every (α, ω)-harmonic map into M . This isthe content of the following theorem:

Theorem 3.6.1. If 1 < α < 3/2, any critical point f ∈ L2α1 (Σ,M) for Eα,ω is

smooth. Moreover, if ψ ∈ L2k(Σ,RN ), for some k ≥ 3, and β ∈ (0, 1], then any

critical point f ∈ L2α1 (Σ,M) for Eβα,ψ,ω is L2

k+2.

The proof given by Sacks and Uhlenbeck [68] relies on results from Morrey[54], a classic book which contains proofs of many such regularity theorems. Acomplete proof of this result goes beyond the scope of these notes. However,we hope the following outline will be helpful in giving an idea as to what isinvolved. Some readers may wish to skip the following sketch, which makes useof the Holder spaces described in §5.1 of [19].

Following [68], we divide the proof into three steps: First we show that thecritical point f is in L2

2, then in a Holder space C1,β , and finally, we use theSchauder theory [25] together with elliptic bootstrapping to prove that f isC∞. We will deal only with Eα,ω, the modification necessary for Eβα,ψ,ω beingrelatively straightforward.

138

Step 1. We show that f lies in L22(Σ,M). To do this, we can use regularity

results for variational problems presented in Evans [19], §8.3.Regularity is a local condition, and we need only show that a critical point f

is regular near a given point p ∈ Σ. Let U and V be small open neighborhoods ofp with V ⊂ U . Our strategy is to work in local coordinates (x1, . . . , xn) definedon a neighborhood of F (U) in M and a local conformal parameter (u1, u2) onU ⊂ Σ. The coordinate system on M allows us to regard f |U as an Rn-valuedmap. We abbreviate the composition xi f to xi, and let η =

∑ηabdu

adub andg =

∑gijdx

idxj denote the Riemannian metrics on Σ and M respectively. Let

pia =∂xi

∂uaso |df |2 =

2∑a,b=1

n∑i,j=1

gij(x1, . . . xn)ηabpiapjb.

Then

Eα,ω(f |U) =12

∫U

L(p, x, u)du1du2,

where L(p, x, u) = (1 + |df |2)α√

det(ηab).

The fact that f |V is (α, ω)-harmonic is expressed by the Euler-Lagrange condi-tion ∫

U

∑a,i

∂L∂pia

∂

∂ua(ζ2vi) +

∑i

∂L∂xi

ζ2vi

du1du2 = 0, (3.38)

for every smooth test function v = (v1, . . . , vn), ζ : Σ → [0, 1] being a suitablesmooth cutoff which is one on V and zero outside U .

Note that the first derivatives of the Lagrangian L are given by

∂L∂pia

= α(1 + |df |2)α−1 ∂

∂pia(|df |2),

where∂

∂pia(|df |2) =

2∑b=1

n∑j=1

gij(x1, . . . xn)ηabpjb√

det(ηab).

Similarly, the second derivatives of L are given by

∂2L∂pia∂p

jb

= α(1 + |df |2)α−1 ∂2

∂pia∂pjb

(|df |2)

+ α(α− 1)(1 + |df |2)α−2 ∂

∂pia(|df |2)

∂

∂pjb(|df |2),

where

∂2

∂pia∂pjb

(|df |2) = gij(x1, . . . xn)ηab√

det(ηab)∂2

∂pia∂xk

(|df |2).

139

Note that by scaling the coordinates appropriately, we can make |∂gij/∂xk| and|∂2gij/∂x

k∂xl| less than ε for any given ε > 0. As a consequence we find thatthe Euler-Lagrange operator for the (α, ω)-energy is uniformly elliptic:

2∑a,b=1

n∑i,j=1

∂2L∂pia∂p

jb

ξiaξjb ≥ αγ

22∑

a,b=1

n∑i,j=1

gij(x1, . . . xn)ηab√

det(ηab)ξiaξjb ,

where γ2 = (1 + |df |2)α−1. It is exactly this estimate on uniform ellipticity thatallows us to apply the difference quotient approach described in Evans [19] toshow that u|V is in L2

2. Since Σ can be covered by finitely many neighborhoodsV , it follows that f ∈ L2

2(Σ,M), finishing Step 1.We remark that it is uniform ellipticity that fails for the function Eβα,ψ,ω when

β = 0 and this explains why critical points of this function are not necessariysmooth.

Step 2. For the second step, we recall that equation (3.35) for α-harmonicmaps can be written in the form

(∆f)> = −(α− 1)d(log(1 + |df |2)) · df,

where (∆f)> =Df

∂x1

(∂f

∂x1

)+Df

∂x2

(∂f

∂x2

),

D denoting the Levi-Civita connection on M . Since we know by Step 1 that f isL2

2, we are justified in differentiating on the right-hand side, thereby obtainingthe following result after a short calculation:

∂2f

∂x21

+∂2f

∂x22

= α(f)(df, df)− (α− 1)B(d2f, df)1 + |df |2

df,

where α(f) and B are bilinear maps, α(f) being the familiar second fundamentalform of M in RN . We can put the last term on the left-hand side obtainingL(f) = α(f)(df, df), where

L(u) =∂2u

∂x21

+∂2u

∂x22

+ (α− 1)B(d2u, df)1 + |df |2

df,

L being a second-order differential operator with coefficients in L∞. Moreover,since α − 1 < 1/2, the operator L is uniformly elliptic. The linear operator Ldefines a bounded linear map from L4

2 to L4, which can be regarded as a smallperturbation of the scalar Laplace operator when α is close to one.

If we restrict L to a small disk D about a given point p and u ∈ Ker(L),each component of u will assume its maximum value on the boundary ∂D ofD. Thus if we impose Dirichlet boundary conditions that u and f agree on ∂D,the map L : L4

2 → L4 will be injective. On the other hand, ∆ : L42 → L4 has

a continuous inverse G0 : L4 → L42, and we can think of G0 as an approximate

inverse to L when α − 1 is small. Given h ∈ L4, then for appropriate choicesof norms on L4

2 and L4, the map u 7→ G0(Lu − h) + u is a contraction, and it

140

must therefore possess a unique fixed point. This implies that L is surjective,and the open mapping theorem implies that the inverse G to L is continuous.

Since the critical point f is continuous and df ∈ L2, α(f)(df, df) is in L4,and there is a unique u ∈ L4

2(D,M) satisfying the Dirichlet boundary conditionssuch that L(u) = α(f)(df, df). This u must of course be f and hence f ∈ L4

2.We thus conclude that the restriction of f to a small neighborhood of any pointlies in L4

2. Since L42 is contained in the Holder space C1,β for suitable β, we

conclude that f ⊂ C1,β(Σ,M).

Step 3. The final step is via the technique of ”elliptic bootstrapping” to thedifferential equation

L(f) = α(f)(df, df), (3.39)

using the Schauder theory. The necessary Schauder estimates are described in[25], Chapter 8 for scalar operators, the general case being a standard extensionwithin PDE theory. The result is that:

f ∈ C1,β ⇒ right side of (3.39) ∈ C0,β ⇒ f ∈ C2,β

⇒ right side of (3.39) ∈ C1,β ⇒ f ∈ C3,β ⇒ · · · .

This shows that f is C∞ and completes our sketch of the argument for thetheorem.

3.7 Morse theory for the perturbed energy

Once we have condition C, we can apply Liusternik-Schnirelmann theory to thefunctions

Eα,ψ,ω(f) =12

∫Σ

(1 + |df |2)αdA+∫

Σ

f · ψdA,

but we would like to establish Morse inequalities and define a Morse-Wittencomplex when the critical points are Morse nondegenerate. This requires anextension of Morse theory to certain functions on Banach manifolds, and such atheory was in fact developed by Uhlenbeck [80]. In this section, we present theresults of that theory, specialized to the perturbations of the energy we havebeen studying.

Note that if f is a critical point for Eβα,ψ,ω, then a calculation similar to thatwhich yields (3.32) shows that

dEα,ψ,ω(f)(V ) = α

∫Σ

(1 + |df |2)α−1〈df,DV 〉dA+∫

Σ

ψ · V,

for all V ∈ TfL2α1 (Σ,M), where DV denotes the covariant differential of V with

respect to the Levi-Civita connection on M .We would also like a formula for the second derivative such as that given

in Proposition 2.4.1. In order to state this formula, we define a linear operator

141

K : TfL2α1 (Σ,M) → TfL

2α1 (Σ,M) in terms of the Riemann curvature R of M

by the formula

〈K(V ),W 〉dA =⟨R

(V,∂f

∂u

)∂f

∂u+R

(V,∂f

∂v

)∂f

∂v,W

⟩dudv,

where (u, v) are isothermal coordinates on M .

Proposition 3.7.1. If f is a critical point for Eα,ψ,ω, then

d2Eα,ψ,ω(f)(V,W ) = α

∫Σ

(1 + |df |2)α−1[〈DV,DW 〉 − 〈K(V ),W 〉]dA

+ 2α(α− 1)∫

Σ

(1 + |df |2)α−2〈df,DV 〉〈df,DW 〉dA

+∫

Σ

ψ · α(V,W )dA, (3.40)

for all V,W ∈ TfL2α1 (Σ,M), where α is the second fundamental form of M in

RN .

Although (3.40) may look complicated at first, note that it specializes to (3.40)in the case where the Riemannian manifold M is just Euclidean space andψ = 0. Moreover, we can set α = 1, and obtain the formula for the Hessian ofthe ordinary energy Eω at a critical point:

Corollary 3.7.2. If f is ω-harmonic, then

d2Eω(f)(V,W ) =∫

Σ

[〈DV,DW 〉 − 〈K(V ),W 〉]dA, (3.41)

for all V,W ∈ TfL2α1 (Σ,M).

The proof of Proposition 3.7.1 is quite similar to the proof of Proposition 2.4.1.For simplicity, we assume that ψ = 0. We consider a variation of f which hasits support within a given coordinate chart (U, (x, y)) on Σ. Recall that such avariation is a smooth family of maps t 7→ f(t) in L2

k(Σ,M) with f(0) = f , andlet

α(x, y, t) = f(t)(x, y), V (x, y) =∂α

∂t(x, y, 0) ∈ Tf(x,y)M.

Setting γ(t)2 = (1 + |df(t)|2)α−1, we obtain

d2Eα,ω(f)(V, V ) =d2

dt2(Eω(f(t))

∣∣∣∣t=0

= α

∫Σ

∂

∂t

[γ2

⟨∂α

∂x,D

∂t

∂α

∂x

⟩+ γ2

⟨∂α

∂y,D

∂t

∂α

∂y

⟩]dxdy

∣∣∣∣t=0

142

where D as usual denotes the covariant derivative of the Levi-Civita connectionon the ambient Riemannian manifold M . We can rewrite this as

d2Eα,ω(f)(V, V ) = α

∫Σ

γ2

[⟨D

∂t

∂α

∂x,D

∂t

∂α

∂x

⟩+⟨D

∂t

∂α

∂y,D

∂t

∂α

∂y

⟩+⟨∂α

∂x,D2

∂t2∂α

∂x

⟩+⟨∂α

∂y,D2

∂t2∂α

∂y

⟩]dxdy

∣∣∣∣t=0

+2α(α− 1)∫

Σ

dγ2

dt

[⟨∂α

∂x,D

∂t

∂α

∂x

⟩+⟨∂α

∂y,D

∂t

∂α

∂y

⟩]dxdy

∣∣∣∣t=0

. (3.42)

When we evaluate at t = 0, we find that

dγ2

dt=〈df,DV 〉1 + |df2

,

where V is the variation field, so we can rewrite (3.42) as

d2Eα,ω(f)V, V ) = α

∫Σ

γ2

[⟨DV

∂x,DV

∂x

⟩+⟨DV

∂y,DV

∂y

⟩+⟨∂f

∂x,D

∂t

DV

∂x

⟩+⟨∂f

∂y,D

∂t

DV

∂y

⟩]dxdy

+ 2α(α− 1)∫

Σ

(1 + |df |2)α−2〈df,DV 〉2dA.

Finally, applying the definition of the Riemann-Christoffel curvature tensor R,we obtain

d2Eα,ω(f)(V, V ) =∫

Σ

γ2

[⟨DV

∂x,DV

∂x

⟩+⟨DV

∂y,DV

∂y

⟩+⟨∂f

∂x,R

(V,∂f

∂x

)V

⟩+⟨∂f

∂y,R

(V,∂f

∂y

)V

⟩+⟨∂f

∂x,D

∂x

DV

∂t

⟩+⟨∂f

∂y,D

∂y

DV

∂t

⟩]dxdy

+ 2α(α− 1)∫

Σ

(1 + |df |2)α−2〈df,DV 〉2dA.

An integration by parts and use of the Euler-Lagrange equation eliminates thethird term, thereby yielding (3.40) in the case where ψ = 0.

If f is a critical point for the function Eα,ψ,ω, we can define a second-orderlinear partial-differential operator

L : L21(f∗TM)→ L2

−1(f∗TM) by d2Eα,ψ,ω(f)(V,W ) =∫

Σ

〈L(V ),W 〉dA,

and we call L the Jacobi operator. The Jacobi operator L restricts to a con-tinuous linear map L : L2

k(f∗TM) → L2k−2(f∗TM) for all k ≥ 2, and if

143

ι : L2k−2(f∗TM) → L2

k(f∗TM) is the inclusion and λ ∈ C, we have an as-sociated operator Lλ = L − λι. The fundamental Theorem 2.4.2 on formallyself-adjoint elliptic operators now applies to the Jacobi operator L, and henceL is a Fredholm operator, which satisfies the additional conditions:

1. for each λ ∈ C the eigenspace Wλ = Ker(Lλ) is finite-dimensional.

2. all the elements of Wλ are C∞,

3. if Wλ is empty, then Lλ possesses a Green’s operator inverse,

4. if Wλ is nonempty, that is, λ is an eigenvalue, then λ ∈ R, and

5. the eigenvalues can be arranged in a sequence

λ1 < λ2 < · · · < λi < · · · with λi →∞,

and only finitely many of the eigenvalues are negative.

On the other hand, if f is a critical point for Eα,ψ,ω, we can also define aninner product

〈〈·, ·〉〉 : TfL2α1 (Σ,M)× TfL2α

1 (Σ,M) −→ R

by the somewhat complicated formula,

〈〈V,W 〉〉 = α

∫Σ

(1 + |df |2)α−1[〈DV,DW 〉+ 〈V,W 〉]dA

+ 2α(α− 1)∫

Σ

(1 + |df |2)α−2〈df,DV 〉〈df,DW 〉dA,

which is useful because it so close to (3.41). Since f is a smooth function, thisis actually equivalent to the simpler L2

1 inner product 〈〈·, ·〉〉0 defined by

〈〈V,W 〉〉0 =∫

Σ

[〈DV,DW 〉+ 〈V,W 〉]dA.

We can then define a second-order linear partial-differential operator

J : L21(f∗TM)→ L2

−1(f∗TM) by 〈〈V,W 〉〉 =∫

Σ

〈J(V ),W 〉dA.

Once again J restricts to a Fredholm operator J : L2k(f∗TM) → L2

k−2(f∗TM)for all k ≥ 2. However, since 〈〈·, ·〉〉 is positive-definite, the eigenvalues of Jare all positive and J itself has a Green’s operator inverse. It is here that thecomplicated formula for the L2

1-inner product on L21(f∗TM) pays off, because

L − J is of zero order, and hence L − J : L2k(f∗TM) → L2

k−2(f∗TM) is acompact operator, for all k ≥ 1.

144

In parallel with the theory of smooth closed geodesics, we can define a con-tinuous linear map

A : L21(f∗TM)→ L2

1(f∗TM) by 〈〈AV,W 〉〉 = d2Eα,ψ,ω(f)(V,W ). (3.43)

Then the operator

A = J−1 L : L21(f∗TM)→ L2

1(f∗TM)

is Fredholm, and restricts to a Fredholm operator on each L2k(f∗TM), for k ≥ 2.

By the Lp theory for elliptic operators, it also restricts to a Fredholm operatoron each Lpk(f∗TM), for p ≥ 2 and k ≥ 2.

But if we let p = 2α and choose q so that (1/p)+(1/q) = 1, then we can alsodefine the operator A directly on Lp1(f∗TM) by (E:explicitformulaA). Indeed,any V ∈ Lp1(f∗TM) defines a continuous linear functional

W 7→ d2E′α,ω(f)(V,W ), for W ∈ Lq1(f∗TM),

and by the duality between Lp1 and Lq1 this linear functional defines an elementAV ∈ Lp1(f∗TM). This defines a continuous linear map

A : TfL2α1 (Σ,M) −→ TfL

2α1 (Σ,M).

such thatA− id : TfL2α

1 (Σ,M) −→ TfL2α1 (Σ,M)

is a compact operator, and thus A is a Fredholm operator as before.

Definition. We say that the critical point f for Eα,ψ,ω is Morse nondegenerateif A is an isomorphism. The Morse index of f is the sum of the dimensions ofthe eigenspaces of L for negative eigenvalues. Finally, Eα,ψ,ω : L2α

1 (Σ,M)→ Ris a Morse function if all of its critical points are Morse nondegenerate.

As in the theory of smooth closed geodesics, we can choose ψ so that Eα,ψ,ω isa Morse function:

Theorem 3.7.3. For a residual set of ψ ∈ L2k(Σ,RN ), the function Eα,ψ,ω is a

Morse function.

The argument is virtually identical to the argument we presented for Theo-rem 2.7.1.

As in the theory of smooth closed geodesics, we can set

M0 = f ∈ L2α1 (Σ,M) : f is constant ,

and then M0 is a nondegenerate critical submanifold for Eα,ω of Morse indexzero. Just as as at the end of §2.7, we can perturb Eα,ω to a function E′α,ω :L2α

1 (Σ,M)→ R such that

1. E′α,ω = Eα,ω on a neighborhood U of M0,

145

2. E′α,ω = Eα,ψ,ω outside a larger neighborhood V of M0, for some ψ ∈L2k(Σ,RN ), and

3. all critical points of E′α,ω either lie within M0 or are Morse nondegenerateand do not lie in V .

The Handle Addition Theorem 2.9.1 can now be modified so that it appliesto the Morse function E′α,ω, and hence the Morse inequalities of §2.10 can beextended to E′α,ω. Let M = L2α

1 (Σ,M) with its standard Finsler metric, andlet

Ma = f ∈ L2α1 (Σ,M) : E′α,ω(f) ≤ a.

Theorem 3.7.4. If the interval [a, b] contains a single critical value c for E′α,ω,there is exactly one critical point p for E′α,ω such that E′α,ω(p) = c and this

critical point is Morse nondegenerate of Morse index λ, then Mb is homotopyequivalent to Ma with a handle of index λ attached.

Indeed, the proof presented in §2.9 was designed precisely so that it would applyto this case. The main difference is that now the model space E is a Banachspace instead of a Hilbert space. and one must make some small modificationsto the argument to account for this.

Moreover, the notion of gradient-like vector field presented in §2.9 was designedso that it would apply directly to the function E′α,ω : L2α

1 (Σ,M) → R. There-fore, we can define stable and unstable manifolds of nondegenerate critical pointsjust as we did before.

Suppose that E′α,ω is a perturbation of Eα,ω which agrees with Eα,ω in aneighborhood of M0 and satisfies the condition that all its critical points areeither in M0 or are Morse nondegenerate. If X is a gradient-like vector fieldfor E′α,ω, we let Ck(E′α,ω,X ) denote the free Z-module generated by the criticalpoints fk,1, fk,2 . . . for E′α,ω of index k and let ∂ be the Z-module homomorphism

∂ : Ck(E′α,ω,X ) −→ Ck−1(E′α,ω,X )

defined by∂(fk,j) =

∑q

ajqfk−1,q, (3.44)

where ajq ∈ Z is the oriented number of trajectories from fk,j to fk−1,q, theorientation being determined as in §2.11.

Theorem 3.7.5. The Z-module homomorphisms thus defined satisfy the iden-tity ∂ ∂ = 0 and the resulting Morse-Witten complex

· · · → Ck+1(E′α,ω,X )→ Ck(E′α,ω,X )→ Ck−1(E′α,ω,X )→ · · ·

calculates the homology of the pair (C0(Σ,M),M0).

The proof is a straightforward modification of the argument presented in §2.11.

146

3.8 Local control of energy density

Our next goal is to investigate the limit of α-energy critical points as α→ 1. Thekey technique here is the Bochner Lemma, which gives a uniform C1 estimateon harmonic and α-harmonic maps whose total energy is small (less than somepositive constant ε0). This in turn is crucial in understanding a key feature ofharmonic and α-harmonic maps, the phenomenon of bubbling. Throughout thissection, we use the notation

e(f) =12|df |2 =

12

(∣∣∣∣∂f∂u∣∣∣∣2 +

∣∣∣∣∂f∂v∣∣∣∣2)dudv,

for isothermal parameters (u, v) on Σ.

Lemma 3.8.1. Suppose that f : D1 → M is a harmonic map, where D1 isthe unit disk in the complex plane, with the standard Euclidean metric ds2 andM is a compact Riemannian manifold with sectional curvatures satisfying theinequality K(σ) ≤ 1. Then for any r0 > 0,∫

D1

e(f)dA < π(r20 − 8r4

0) ⇒ maxσ∈(0,1]

σ2 supD1−σ

e(f) < 4r20. (3.45)

Proof: Choose σ0 ∈ (0, 1] so that

σ20 supD1−σ0

e(f) ≥ σ2 supD1−σ

e(f), for all σ ∈ (0, 1],

and choose p0 ∈ D1−σ0 so that

e0 = e(f)(p0) = supe(f)(p) : p ∈ D1−σ0.

If σ20e0 < 4r2

0, the lemma is proven. So it suffices to assume that σ20e0 ≥ 4r2

0,and derive a contradiction. But then(σ0

2

)2

supD1−σ0/2

e(f) ≤ σ20 supD1−σ0

e(f) = σ20e0 ⇒ sup

D1−σ0/2

e(f) ≤ 4e0.

Moreover, σ20e0 ≥ 4r2

0 implies that

r0√e0

≤ σ0

2,

so we can define a new harmonic map g : Dr0 →M by

g(q) = f

(p0 +

q√e0

)such that e(g)(0) = 1 and e(g) ≤ 4.

It now follows from (3.28), which follows from the Bochner Lemma, that

∆(e(g)) ≥ −2(e(g))2 ≥ −32.

147

Thus if we define a smooth function h : Dr0 → R in terms of the standardcoordinates (u, v) by

h = e(g)− [1− 16(u2 + v2)],

then we find that∆h = ∆(e(g)) + 32 ≥ 0,

and hence by the maximum principle,

0 = h(0) ≤∫Dr0

hdA ≤∫Dr0

[e(g)− (1− 16(u2 + v2))]dA

≤∫Dr0

e(g)dA−∫ 2π

0

∫ r0

0

(1− 16r2)rdrdθ,

and an integration yields the inequality∫D1

e(f)dA > π(r20 − 8r4

0).

Thus we have proven the contrapositive of the assertion in the lemma.

Corollary 3.8.2. Suppose that f : D1 →M is a harmonic map. Then∫D1

e(f)dA <π

16⇒ max

σ∈(0,1]σ2 sup

D1−σ

e(f) <8π

∫D1

e(f)dA.

To prove this we note that if r0 ≤ (1/4),∫D1

e(f)dA =π

2(r2

0) ≤ π(r20 − 8r4

0)

⇒ maxσ∈(0,1]

σ2 supD1−σ

e(f) < 4r20 =

8π

∫D1

e(f)dA.

Corollary 3.8.3. Suppose that Dr is the disk of radius r in the complex planeand f : Dr →M is a harmonic map. Then∫

Dr

e(f)dA <π

16⇒ max

σ∈(0,r]σ2 sup

Dr−σ

e(f) <8π

∫Dr

e(f)dA.

To prove this, simply note that under rescaling from D1 to Dr, the energyintegral remains unchanged, while

|df | 7→ 1r|df |, e(f) 7→ 1

r2e(f)

and

maxσ∈(0,1]

σ2 supD1−σ

e(f) 7→ maxσ∈(0,1]

(σr

)2

supDr−rσ

e(f) = maxσ∈(0,r]

τ2 supDr−τ

e(f).

148

Of course, we would like versions of the previous lemmas for the α-energy Eα,ωwhen α is sufficiently close to one.

Lemma 3.8.4. Suppose that M is a compact Riemannian manifold whosesectional curvatures satisfy the bound K(σ) ≤ 1. There exists α0 ∈ (1,∞) andfor each α ∈ [1, α0] a homogeneous second-order elliptic operator Lα such thatsuch that the coefficients of Lα tend uniformly to those of ∆ as α→ 1, and

Lα(e(f)) ≥ [2Ke(f)− 2e(f)2],

where K is the Gaussian curvature of the Riemannian metric on Σ.

Proof: It follows from (3.21) and the discussion leading to (3.28) that

∆η(e(f)) = |∇df |2 + 〈d[(∆ηf)>], df〉+ 2Ke(f)− 2(e(f))2.

Recall that equation (3.35) for α-harmonic maps can be written in the form

(∆ηf)> = −(α− 1)d(log γ2) · df, where γ2 = 1 + |df |2,

Since

d(log γ2) =d(|df |2)1 + |df |2

=2〈∇df, df〉1 + |df |2

,

∇d(log γ2) =∇d(|df |2)(1 + |df |2) + (d(|df |2))2

(1 + |df |2)2=∇d(|df |2)(1 + |df |2)

+4〈∇df, df〉2

(1 + |df |2)2,

we conclude that there is a homogeneous second-order differential operator Q(with uniformly bounded coefficients which depend on df) such that

|〈∇(∆f), df〉| ≤ (α− 1)Q(e(f)) + (α− 1)(constant)〈∇df, df〉2

(1 + |df |2)2,

the constant being independent of f and α. The second term can be absorbedin |∇df |2 when α is sufficiently close to one, while the first term can be absorbedin the elliptic operator Lα.

One can now use Lemma 3.8.4 in place of the standard Bochner Lemma to provea version of Lemma 3.8.3 for α-harmonic maps.

Lemma 3.8.5. Suppose that M is a compact Riemannian manifold whosesectional curvatures satisfy the bound K(σ) ≤ 1 and that Dr is the disk ofradius r in the complex plane, with the standard Euclidean metric ds2. Thereexists an α0 > 1 and an ε0 > 0 with the following property: If f : Dr → M isan α-harmonic map,∫

Dr

e(f)dA < ε0 ⇒ maxσ∈(0,r]

σ2 supDr−σ

e(f) < 3∫Dr

e(f)dA. (3.46)

149

The proof is a straightforward modification of the proof of Lemmas 3.8.1 to3.8.3, in which we use Lemma 3.8.4 for the elliptic operator Lα instead of theordinary Bochner Lemma for the Laplace operator ∆η.

We can next relax the condition that the Riemannian metric on Σ be flat andrescale the metric on M so that it satisfies the condition K(Σ) ≤ K0 for K0 > 0:

Lemma 3.8.6. Suppose that M is a compact Riemannian manifold whosesectional curvatures satisfy the bound K(σ) ≤ K0 where K0 > 0 and that Dr

is the disk of radius r in the complex plane, a Riemannian metric ds2 which isrelated to the flat metric ds2

0 by inequalities of the form

1L2ds2

0 ≤ ds2 ≤ L2ds20,

L being a constant. There exists an α0 > 1 and constants ε0 > 0 and CLdepending on L such that: If f : Dr → M is an α-harmonic map with α ∈(1, α0],∫

Dr

e(f)dA <ε0√K0

⇒ maxσ∈(0,r]

σ2 supDr−σ

e(f) < CL

∫Dr

e(f)dA. (3.47)

Proof: The estimates (3.46) holds for the energy density e0(f) with respect tothe flat metric ds2

0. Now simply use the inequalities

1L2e0(f) ≤ e(f) ≤ L2e0(f)

to obtain (3.47) for K0 = 1. Now note that multiplying the metric by√K0 also

multiplies the energy density by the same factor.

Inequality (3.47) is typically used as follows: Let Σ be a compact Riemannsurface and M a compact Riemannian manifold, r0 > 0 a fixed radius. Thenthere are constants ε0 > 0, C > 0 and α ∈ [1, α0] which depends upon Σ, Mand r0 such that if Dr is a disk of radius r ≤ r0 in Σ and f : Dr → M is anα-harmonic map,∫

Dr

e(f)dA < ε0 ⇒ supDr/2

e(f) <C

r2

∫Dr

e(f)dA. (3.48)

In other words, if the energy on a ball of small radius is sufficiently small, itgives a bound on the energy density itself on a ball of half the radius.

Theorem 3.8.7. Suppose that M is a compact Riemannian manifold and Σis a closed connected Riemann surface. If ωm is a sequence of conformalstructures ∈ T such that ωm → ω∞ ∈ T and fm : Σ → M is a sequence of(αm, ωm)-harmonic maps such that αm → 1 and E(fm) ≤ E0 for some constantE0 > 0, then there is a finite collection of points

p1, . . . , pl ⊆ Σ,

150

and a subsequence of fm (which we still denote by fm), such that fmconverges uniformly in Ck on compact subsets of Σ− p1, . . . , pl, for any non-negative integer k, to an ω∞-harmonic map

f∞ : Σ− p1, . . . , pl −→M.

Proof: We begin by choosing ε0 > 0 so that when r ≤ r0, (3.48) implies that∫Dr

e(f)dA < ε0 ⇒ supDr/2

e(f) <Cε0r2

. (3.49)

Let rm = (1/2)m, for each positive integer m and let Om be an open cover ofΣ by disks Drm(pi), so that each each point of Σ is covered by at most h disksand the disks Drm/2(pi) still cover M . (We can choose h to be independent ofm.) Then ∑

i

∫Drm (pi)

e(fm) ≤ hE0.

Hence there are at most hE0/ε0 of the disks in the cover Om on which∫Drm (xi)

e(fm) ≥ ε0.

For each j we let p1m, . . . , plm be the center points of these disks. Afterpossibly passing to a subsequence, we can arrange that

p1m → p1, . . . , plm → pl.

Then the fm’s are uniformly bounded and equicontinuous on compact subsetsof Σ − p1, . . . , pl. By Arzela’s Theorem the fm’s converge to a continuousfunction f∞ on Σ − p1, . . . , pl, and the convergence is uniform on compactsubsets.

To finish the proof, we use the process of “elliptic bootstrapping” to showthat fm → f∞ in Ck on Σ−p1, . . . , pl, uniformly on compact subsets, for anyk ≥ 0. Indeed, if M is isometrically imbedded in EN , the equation for which fmust satisfy to be an α-harmonic map is

∂2f

∂x2+∂2f

∂y2= −α(f)(df, df)− (α− 1)

B(d2f, df)1 + |df |2

df, (3.50)

where α(f) and B are bilinear maps, α(f) being the familiar second fundamentalform. It follows from (3.49) that dfm is bounded in Lp for all p on any diskwithin Σ−p1, . . . pl. It then follows from multiplication theorems for Sobolevspaces that the right-hand side of (3.50) is bounded in Lp for all p on any suchdisk, and hence elliptic estimates give that f is bounded in Lp2 for all p. Butnow one of the Sobolev imbedding theorems implies that a subsequence of fm

151

converges in C1,β for some β > 0 on any compact subset of Σ−p1, . . . pl. Nowwe can apply Schauder estimates to see that a subsequence of fm convergesin C2,β , then in C3,β , and so forth. Finally we obtain a subsequence of fmwhich converges uniformly in every Ck on compact subsets of Σ− p1, . . . , pl,and the theorem is proven.

Theorem 3.8.8. Suppose that M is a compact Riemannian manifold withnonpositive sectional curvatures. If ω is a conformal structure on the compactRiemann surface Σ, there is an ω-energy minimizing harmonic map in any freehomotopy class [Σ,M ].

In this case, Lemma 3.8.6 implies that for some α0 ∈ (1,∞) and some r0 ≥ 0,whenever Dr is a disk of radius r ≤ r0 in Σ and f : Dr →M is an α-harmonicmap with α ∈ (1, α0],

supDr/2

e(f) < 4CL∫Dr

e(f)dA. (3.51)

Thus we do not need the complicated covering argument utilized in the proofof Theorem 3.8.7.

Instead, we simply take a sequence αm → 1 and for each αm choose anαm-harmonic map fm : Σ→ M which lies in a given component of [Σ,M ] andminimizes the αm-energy on that component. Then (3.51) implies that e(fm)is bounded on any ball Dr/2, which implies that fm converges uniformly oncompact subsets. Elliptic bootstrapping then implies that the sequence fmconverges uniformly in all Ck, proving the theorem.

Theorem 3.8.8 is just the Eells-Sampson Theorem 3.1.1 in the special case whereΣ has dimension two. Similarly, we can prove Hartman’s Theorem 3.1.2 in thespecial case where Σ is a surface:

Theorem 3.8.9. Suppose that M is a compact Riemannian manifold withnegative sectional curvatures. If (Σ, ω) is any Riemann surface, there is at mostone ω-harmonic map in any free homotopy class [Σ,M ].

Indeed, the corresponding statement for (α, ω)-harmonic maps follows from thefact that since ⟨

R

(V,∂f

∂u

)∂f

∂u+R

(V,∂f

∂v

)∂f

∂v, V

⟩≤ 0,

any (α, ω)-harmonic map is stable by (3.40). If there were two (α, ω)-harmonicmaps in the same component of C0(Σ,M), there would have to be an unstablecritical point by the Morse inequalities, which cannot happen.

To prove the corresponding statement for the limiting case of ω-harmonicmaps, we can use the implicit function theorem. For this, we define a smoothmap

E?ω : L2k(Σ,M)× [1, α0) −→ R by E?ω(f, α) = Eα,ω(f),

152

and observe that at α = 1 we obtain the usual energy up to a constant,E?ω(f, 1) = Eω(f) + (1/2). We can differentiate to obtain the Euler-Lagrangemap

F ∗ω : L2k(Σ,M)× [1, α0) −→ L2

k−2(Σ, TM).

If f is an (α, ω)-harmonic map, we can compose the differential DF ∗ω of F ∗ω at(f, α) with the projection onto the vertical obtaining the corresponding Jacobioperator,

L = πV DF ∗ω(f, α) : TfL2k(Σ,M)⊕ R −→ TfL

2k−2(Σ,M).

Note that L is a Fredholm operator with Fredholm index one, and that f is aMorse nondegenerate critical point for Eα,ω if and only if L is surjective.

Suppose now that f : Σ → M is an ω-conformal harmonic map, and notethat f must be Morse nondegenerate by Corollary 3.7.2. It follows from theimplicit function theorem that if U is a sufficiently small C2 neighborhood of(f, 1) in L2

k(Σ,M)× [1, α0), then

(F ∗ω)−1(zero section of T (Map(Σ,M)))

is a smooth submanifold of U of dimension one. This submanifold consists ofcritical points of Eα,ω for α varying in some interval [1, α0) for which the corre-sponding Jacobi operators L are surjective. All the points of this submanifoldare Morse nondegenerate critical points for Eα,ω.

Thus two distinct ω-harmonic maps f, g : Σ → M in the same componentof C0(Σ,M) would give rise to two distinct one-parameter families fα, gα of(α, ω)-harmonic maps in the same component of C0(Σ,M), contradicting theuniqueness of (α, ω)-harmonic maps. This finishes the proof of Theorem 3.8.9.

3.9 Bubbling

In §3.7 we have shown existence of many α-energy critical points, correspondingto numerous topological constraints, while in §3.8 we have shown that given asequence fm : Σ→M of αm-harmonic maps with αm > 1 such that αm → 1and E(fm) ≤ E0 for some constant E0 > 0, there is a collection of pointsp1, . . . , pl and a subsequence of fm (which we still denote by fm) whichconverges uniformly on compact subsets of Σ − p1, . . . , pl, together with thefirst k derivatives to a map

f∞ : Σ− p1, . . . , pl −→M, (3.52)

which is harmonic on Σ− p1, . . . , pl. The following theorem implies that f∞extends to a harmonic map defined on the entire Riemann surface Σ:

Removeable Singularity Theorem 3.9.1. Let D be the unit disk in thecomplex plane C. If

f : D − 0 −→M

153

is a harmonic map of finite energy, then f extends to a smooth harmonic mapon the entire disk D.

We apologize to the reader for not giving a proof of this theorem in these notes.It is proven as Theorem 3.6 in [68].

Returning now to our sequence fm of αm-harmonic maps with αm → 1and E(fm) ≤ E0, we note that as measures, the sequence of Radon measurese(fm)dA converges weakly to

e(f∞)dA+∑i=1l

miδpidA,

where mi is a nonnegative constant and δpi is the Dirac delta-function located atpi. If mi is zero, we can throw the point pi away, since in this case Lemma 3.8.6implies that after possibly passing to a subsequence fm will converge on aneighborhood or pi. If mi > 0 is nonzero, we call pi a bubble point of thesequence and mi measures the amount of energy lost at the bubble point.

We next investigate what happens at the bubble points p1, . . . , pl. For eachm, choose qim ∈ Drm(pim) so that

e(fm)(qim) = supe(fm)(q) : q ∈ Drm(pim).

Note that qim → xi as j approaches ∞, and let

bim =√e(fm)(qim).

If no subsequence of the fim’s converge near pi then bim → ∞. Assume thatthe latter alternative holds, and define

fm : Drmbim(0)→M by fm(q) = fm

(qim +

p

bim

).

One readily verifies that e(fim) ≤ 1 and e(fim)(0) = 1. Therefore a subsequenceof the fm’s converges to a nonconstant harmonic map

f∞ : C− q1, . . . qm −→M,

the convergence being uniform in Ck for all k on compact subsets of C −q1, . . . qm. At the finitely many points q1, . . . qm, further bubbling can occur.

Recall that as a Riemann surface the complex plane is just the Riemannsphere S2 minus a point. Therefore the removable singularity theorem impliesthat f∞ extends to a smooth harmonic map f∞ : S2 →M .

It is clearly of interest to understand what α-energy critical points look likenear bubble points. This problem has been studied by Parker and Wolfson [64],[63] and Chen and Tian [12], among others.

Remark 3.9.2. Each harmonic two-sphere which bubbles off carries with it acertain minimal amount of energy. Indeed, a harmonic two-sphere h : S2 →M

154

is automatically conformal and hence a minimal surface, and from the Gaussequation, its Gaussian curvature K : S2 → R is bounded above by K0, themaximum value of all sectional curvatures on M . Hence by the Gauss-Bonnettheorem,

4π =∫S2KdA ≤ K0(Area of h(S2)) ⇒ E(h) ≥ 4π

K0.

In particular, if the energy of a sequence of αm-harmonic maps is bounded byE0, the number of harmonic two-spheres that can bubble off in the limit is≤ (4πE0)/K0.

We can now prove a theorem due to Sacks and Uhlenbeck [68], which can bethought of as the analog within minimal surface theory of the Fet-LusternikTheorem 2.3.2 on existence of smooth closed geodesics:

Theorem 3.9.3. If M is a compact simply connected Riemannian manifold,then M contains at least one nonconstant minimal two-sphere.

Proof: In many ways, the argument is very similar to that given in §2.3. Thereis a least integer q ≥ 2 such that Hq(M ; Z) 6= 0, and it follows from the Hurewicztheorem that

πi(M) = 0, for 0 < i < q, and πq(M) ∼= Hq(M,Z) 6= 0.

Let M be the space of smooth maps from S2 to M and let π :M→M byπ(f) = f(0), the evaluation of f at the basepoint 0 ∈ S2 = C∪∞. It is a well-known fact from homotopy theory that π is a fibration with fiberMp = π−1(p),the set of basepoint preserving maps from S2 to M . Moreover,

πk(Mp) ∼= πk+2(M)

and the fibration π induces a long exact sequence

· · · → πk(Mp)→ πk(M)→ πk(M)→ πk−1(Mp)→ · · · .

We note, moreover, that π∗ : πk(M)→ πk(M) possesses a right inverse

i∗ : πk(M)→ πk(M) induced by the inclusion i : M →M

which takes a point to the constant map at that point. Hence the long exactsequence splits and we conclude that

πk(M) ∼= πk(M)⊕ πk(Mp) ∼= πk(M)⊕ πk+2(M).

Thus the homotopy groups of M are completely determined by the homotopygroups of M . In particular,

πq−2(M) ∼= πq(M) 6= 0.

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Since M is simply connected, q ≥ 2 and πq−1(M) ∼= πq(M) is abelian.Moreover, we can identify πq−2(M) with [Sq−2,M], the space of free homotopyclasses of maps from Sq−2 into M. Choose a nonzero element α ∈ [Sq−2,M].Let

F = g(Sq−2) such that g : Sq−2 →M is a continuous map in [α].

Clearly, F is an ambient isotopy invariant family of sets. Hence Minimax(Eα,F)is a critical value for Eα.

Recall that M is isometrically imbedded in an ambient Euclidean spaceRN and let M(δ) denote the open δ-neighborhood of M in RN for δ > 0.By the tubular neighborhood theorem, if δ is sufficiently small, M is a strongdeformation retract of M(δ). Moreover, for ε > 0 sufficiently small, any mapf : S2 →M of energy < ε can be contracted to its center of mass in RN withoutleaving M(δ).

Suppose now that g(Sq−1) ⊂ J−1([0, ε]). Then g is homotopic to a smoothmap

g : Sq−1 →M0, where M0 = γ ∈M : γ is constant .

Hence Minimax(Eα,F) ≥ ε. We now take a decreasing sequence of real num-bers αm with αm → 1. The Minimax Theorem 1.11.2 gives a correspondingsequence fm of critical points for Eαm which have energy ≥ ε > 0. Eithera subsequence converges to a nonconstant harmonic map or a bubble forms inthe limit. In the latter case, the bubble provides a nonconstant harmonic map.In either case we get a nonconstant harmonic map f∞ : S2 → M which is au-tomatically conformal, since there is only one conformal structure on S2. Theimage is a nonconstant minimal two-sphere, which proves the theorem.

3.10 Existence of minimizing spheres

Let M be a compact manifold and suppose that f : D1 → M is a map withf(∂D1) = p which represents an element [f ] ∈ π2(M,p). Suppose, moreover,that γ : [0, 1]→ M is a smooth map with γ(0) = p = γ(1) which represents anelement [γ] ∈ π1(M,p). We can then define γ ? f as follows: Let (r, θ) be polarcoordinates on D1 and set

g(r, θ) = (γ ? f)(r, θ) =

f(2r, θ), for 0 ≤ r ≤ 1/2,γ(2r − 1), for 1/2 ≤ r ≤ 1.

Then [γ], [f ] 7→ [g] = [γ ? f ] gives an action of π1(M,p) on π2(M,p), whichmakes π2(M,p) into a Z[π1(M,p)]-module, where Z[π1(M,p)] is the group ringof π1(M,p). This action is discussed in more detail in Chapter 4 of [35].

Theorem 3.10.1 (Sacks and Uhlenbeck). Suppose that M is a compactRiemannian manifold. Then a generating set for π2(M,p) as a Z[π1(M,p)]-module can be represented by minimal two-spheres, possibly with branch points.

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Note that the minimal two-spheres need not pass through the point p. If Mis simply connected, it follows from the Hurewicz Theorem that π2(M,p) ∼=H2(M ; Z), and hence we obtain:

Corollary 3.10.2. If M is a compact simply connected Riemannian man-ifold, then a set of generators for H2(M ; Z) can be represented by minimaltwo-spheres.

Before proving Theorem 3.10.1, we make a few remarks regarding the symmetryof the functions E,Eα : C2(S2,M) → R. We regard S2 as the one-pointcompactification C∪∞ of the complex plane C with the standard coordinate

z = x+ iy = reiθ = eu+iθ,

and note that E is invariant under all the linear fractional transformations: Ifφ : S2 → S2 is the diffeomorphism defined by

φ(z) =az + b

cz + d, for

(a bc d

)∈ SL(2,C),

then E(f φ) = E(f). Note that(a bc d

)=(−1 00 −1

)⇒ f φ = f,

so the group of linear fractional transformations is actually

G = PSL(2,C) =SL(2,C)±I

.

In contrast, when α > 1, the maps E0α, Eα : C2(S2,M)→ R defined by

E0α(f) =

12

∫Σ

|df |2αdA and Eα(f) =12

∫Σ

(1 + |df |2)αdA.

are invariant only under the smaller group of isometries SO(3) ⊆ PSL(2,C).Indeed, critical points of E0

α must be parametrized so that the “center of mass”is zero:

Lemma 3.10.3. Let X : S2 → R3 denote the standard inclusion, and let0 denote the origin in R3. If f ∈ C2(S2,M) is a critical point for E0

α withα ∈ (1,∞), then ∫

S2X|df |2αdA = 0.

Moreover, there is a smooth function ψα : [0,∞) → R such that ψα(0) = 0,ψα(t), ψ′α(t) > 0 for t > 0, and if f is a critical point for Eα, then∫

S2Xψα(|df |2)dA = 0. (3.53)

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In the argument for this lemma, we utilize the standard metric on S2 of constantcurvature one, expressed in terms of our standard coordinates by

ds2 =4

(1 + r2)2(dr2 + r2dθ2) =

1cosh2 u

(du2 + dθ2),

where r = e−u. (This metric is related by scaling to the constant curvaturemetric of total area one.) For t ∈ R, we define a linear fractional transformation

φt : S2 → S2 by u φt = u+ t, θ φt = θ,

so that t 7→ φt is a one-parameter subgroup. The energy density is then givenby the formula

e(f φt) =12

(∣∣∣∣∂f∂u∣∣∣∣2 +

∣∣∣∣∂f∂θ∣∣∣∣2)2

cosh2(u+ t),

and it is straightforward to calculate its derivative at t = 0:

d

dte(f φt)

∣∣∣∣t=0

=12d

dt|d(f φt)|2

∣∣∣∣t=0

=

(∣∣∣∣∂f∂u∣∣∣∣2 +

∣∣∣∣∂f∂θ∣∣∣∣2)

sinhu coshu = |df |2 tanhu. (3.54)

Using this fact, we can perform a straightforward calculation to obtain

d

dtE0α(f φt)

∣∣∣∣t=0

= (α− 1)∫S2|df |2α(tanhu)dA.

In terms of the standard Euclidean coordinates (x, y, z) on R3, S2 is representedby the equation x2 + y2 + z2 = 1, and a straightforward computation usingstereographic projection from the north pole to the (x, y)-plane shows that

z =r2 − 1r2 + 1

=sinhucoshu

= tanhu,

so if f is a critical point for E0α,

0 =d

dtE0α(f φt)

∣∣∣∣t=0

= (α− 1)∫S2|df |2αzdA.

But we can take the z-axis to be any line passing through the origin, and hence∫S2

X|df |2αdA = 0,

for the metric of constant curvature one, and hence for the constant curvaturemetric of total area one.

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More generally, we can use (3.54) to calculate the derivative of Eα to obtain

d

dtEα(f φt)

∣∣∣∣t=0

=12d

dt

∫§2

(1 + |d(f φt)|2)αsech2(u+ t)dudθ∣∣∣∣t=0

= α

∫S2

(1 + |df |2)α−1 tanhu|df |2dA−∫S2

(1 + |df |2)α tanhudA.

Thus, if f is a critical point for Eα,

0 =∫S2

[α(1 + |df |2)α−1|df |2 − (1 + |df |2)α]zdA,

=∫S2

[α(1 + |df |2)α−1|df |2 − (1 + |df |2)α − 1]zdA,

where we have used the fact that the average value of z on S2 is zero. We obtain(3.53) by setting

ψα(t) =α(1 + t)α−1t− (1 + t)α − 1

α− 1=∫ t

0

ατ

(1 + τ)2−α dτ. (3.55)

One easily verifies that this function has the desired properties, and has a smoothlimit as α→ 1, namely

ψ1(t) =∫ t

0

τ

(1 + τ)dτ. (3.56)

Remark 3.10.4. It is sometimes convenient to demand that harmonic mapsf : S2 →M satisfy the side condition∫

S2Xψ1(|df |2)dA = 0,

thereby reducing the symmetry group of the problem from PSL(2,C) to a max-imal compact subgroup SO(3).

Lemma 3.10.5. Suppose that M is a compact Riemannian manifold and K0 >0 is an upper bound for the sectional curvatures of M and α0 ∈ (1,∞) issufficiently close to one. Then there is an ε0 > 0 such that every nonconstantα-harmonic map f : S2 →M satisfies

Eα(f)− 12≥ E(f) ≥ ε0.

We give S2 the constant curvature metric of total area one, which has Gaussiancurvature K = 2π. Then it follows from Lemma 3.8.4 that there is an ellipticoperator Lα such that

Lα(e(f)) ≥ 2(2π − e(f))e(f).

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Moreover, it follows from Lemma‘3.8.5 that there is an ε0 > 0 such that

α ∈ (1, α0] and E(f) < ε0 ⇒ e(f) < 2π.

This yields a contradiction to the maximum principle for Lα unless e(f) isidentically zero.

Lemma 3.10.6. Suppose that fm is a sequence of nonconstant critical pointsfor Eαm with αm → 1 such that fm converges to a harmonic map f∞ : S2 →Mon S2 − 0, uniformly on compact subset in Ck for all k. Then f∞ cannot bea map to a point.

Indeed, it follows from the previous lemma that Eα(fm) ≥ ε0 and hence if f∞were constant a nontrivial bubble would form near the point 0 ∈ S2. Thus en-ergy density would have to concentrate at the point 0 and this would contradict(3.53).

Proof of Theorem 3.10.1: The proof is an induction which makes use of a suc-cession of variational problems. Let

F1 = f ∈ C2(S2,M) : the free homotopy class of f is nontrivial ,

and letµ1 = infE(f) : f ∈ F1.

Suppose that αm → 1 and let fm be an element of F1 which achieves a minimumfor Eαm on F1. Either a subsequence of fm converges without bubbling to aminimal two-sphere f∞ : S2 → M which lies within F1, in which case we havea nonconstant minimal two-sphere f ∈ F1 such that E(f1) = µ1, or at least onenonconstant minimal minimal two-sphere h : S2 → M bubbles off at a pointp1 ∈ S2. We need to show that the second possibility leads to a contradiction.

Recall that a subsequence of fm will converge to a harmonic map f∞ :S2 → M , uniformly on compact subsets of S2 − p1, . . . , pl, where p1, . . . , plare the bubble points. By Lemma 3.10.6, either f∞ is nonconstant or there is atleast one other bubble point. Let us suppose that f∞ is nonconstant. (The othercase can be treated in a similar fashion.) We will perform surgeries on smallcircles about p1 to divide each αm-harmonic map into a base map fm : S2 →Mand a bubble hm : S2 →M .

In more detail, let Drm(p1) be a disk of radius rm chosen so that rm → 0 asm→∞,∫

Drm (p1)

e(fm)dA ≥ ε0 and fm(Drm(p1)−D(1/3)rm(p1)) ⊆ Nδ(q)

where q = f∞(p1) and Nδ(q) is the domain of a geodesic coordinate system ofradius δ in M . Then define a map fm : S2 →M by

fm(p) =

fm(p), for p ∈ S2 −Drm(p1),expq(η(p)(expq)−1fm(p)), for p ∈ Drm(p1),

160

where η : S2 → [0, 1] is a smooth function such that

η ≡

1 on S2 −D(2/3)rm(p1),0 on D(1/3)rm(p1),

Let Rm : S2 → S2 be the reflection through the circle ∂D(1/2)rm(p1), and definehm : S2 →M by

hm(p) =

expq(η Rm(p)(expq)−1Rm fm(p)), for p ∈ S2 −Drm(p1),fm(p), for p ∈ Drm(p1),

Thus fm agrees with fm outside Drm(p1) while Rm hm agrees with fm insideDrm(p1).

We can clearly arrange that

Eαm(fm) + Eαm(hm) < Eαm(fm) +ε02

if m is sufficiently large, where ε0 is the constant appearing in Lemma 3.10.5. Atleast one of the maps fm or hm must be homotopically nontrivial with energyless than the infimum over F1, and this provides the desired contradiction.

Thus we obtain a homotopically nontrivial minimal two-sphere f1 : S2 →Mand we let Γ1 denote the Z[π1(M,p)]-submodule of π2(M,p) generated by [f1].If Γ1 6= π2(M,p), we let

F2 = f ∈ C2(S2,M) : the free homotopy class of f is not in Γ1 ,

set µ2 = infE(f) : f ∈ F2, and proceed exactly as before. One therebyobtains a minimal two-sphere f2 : S2 → M which does not lie in Γ1. we thenlet Γ2 be the Z[π1(M,p)]-submodule of π2(M,p) generated by [f1] and [f2],and so forth. For the inductive step, we suppose we have already constructeda Z[π1(M,p)]-submodule Γk−1 of π2(M,p) generated by minimal two-spheres[f1], . . . [fk−1]. If Γk−1 6= π2(M,p), we let

Fk = f ∈ C2(S2,M) : the free homotopy class of f is not in Γk−1 ,

let µk = infE(f) : f ∈ Fk, and verify that there is a nonconstant minimaltwo-sphere fk ∈ Fk such that E(fk) = µk. Theorem 3.10.1 follows by induction.

The above theorem can be applied to compact three-dimensional Riemannianmanifolds. In this case, surfaces which minimize area are free of branch pointsby a theorem of Osserman and Gulliver (see [58] and [31]). In fact the minimalspheres which generate π2(M,p) are either imbedded or double coverings ofprojective planes:

Theorem 3.10.7 (Meeks and Yau). Suppose that M is an oriented com-pact three-dimensional Riemannian manifold. Then there is a finite collectionf1, . . . , fl of generators for π2(M,p) as a Z[π1(M,p)]-module, each of which

161

is either an embedded minimal two-sphere or a doubly covered imbedded pro-jective plane.

We refer the reader to [48] for the proof, which is based upon the tower con-struction of Papakyriakopoulos.

Thus one can divide a compact orientable three-dimensional manifold M alongthe homotopically nontrivial minimal two-spheres which separate M , obtaininga connected sum decomposition

M = M0]M1] · · · ]Mk. (3.57)

Suppose that an element π2(Mi) is nonzero for some i. Then there exists animbedded two sphere N ⊂Mi and two points close to N but on opposite sidescan be connected by an arc C within Mi. A small neighborhood of N ∪ C willhave boundary diffeomorphic to a separating two-sphere. It then follows fromthe positive resolution to the Poincare conjecture that Mi is diffeomorphic toS1 × S2. Thus each summand in (3.57) either has vanishing π2 or is S1 × S2.

A compact orientable three-dimensional manifold is said to be prime if itcannot be expressed as a nontrivial connected sum. It also follows from thePoincare conjecture that each factor in the connected sum decomposition (3.57)is prime, and the decomposition must be the prime decomposition for a compactoriented three-manifold M discovered by Kneser and Milnor. The remarkablefact is that the decomposition occurs along imbedded two-spheres which areminimal with respect to any preassigned Riemannian metric.

Example 3.10.8. Here is an explicit example illustrating how much gets lostof the critical point theory for α-harmonic maps as α→ 1. Starting with a lensspace L(3, 1) of constant curvature one, we consider the connected sum M =L(3, 1)]L(3, 1)]L(3, 1) along isolated embedded minimal two-spheres N1 andN2 of very small radius of curvature which minimize within their free homotopyclasses. From van Kampen’s theorem it follows that the fundamental groupis generated by elements a, b and c, with the relations a3 = b3 = c3 = 1.Explicit construction of the universal cover shows that π2(M ; p) is generated asa π1(M)-module by the two imbedded minimal spheres

f1 : S2 → N1, f2 : S2 → N2.

We can construct an additional embedded two-sphere N = N1]N2 by connectingN1 with N2 by a very thin tube and a corresponding imbedding f : S2 → Nwhich represents the homotopy class [f1] + [f2], which is not freely homotopicto any multiple of [f1] or [f2]. If α0 is sufficiently close to one, then for eachα ∈ (1, α0] there is a minimizing α-minimal two-sphere fα in the component ofMap(S2,M) representing this free homotopy class. As α→ 1 a subsequence ofthese α-minimal two-spheres should approach a configuration consisting of N1,N2 and a minimal geodesic connecting N1 and N2 of some length bounded awayfrom zero.

Although π2(M,p) is generated by two imbedded spheres as a Z[π1(M,p)]-module, it is interesting to note that as an abelian group, π2(M,p) has infinitely

162

many generators, as one sees by noting that it is isomorphic to the secondhomotopy group of its universal cover.

3.11 Existence of minimal tori

Suppose we want to investigate the topology of a compact oriented three-dimensional manifold. We start by using π2(M) to express M as a direct sumdecomposition (3.57) in which each of the summands is prime. One can showthat each prime summand is one of three types: it is diffeomorphic to S1 × S2,it is finitely covered by S3, or all of its homotopy groups are trivial except for]pi1 = π, so it is a K(π, 1).

The next step in exploring the topology is to try to divide up the K(π, 1)summands along tori. An imbedded torus or Klein bottle in M is incompress-ible if the inclusion induces an injection on π1. To construct incompressibletori, we look for subgroups of π1(M) which are isomorphic to Z ⊕ Z and aremaximal among subgroups with this property. Indeed, according to Thurston’sgeometrization program (as described, for example, in [54]), any prime compactoriented three-manifold can be decomposed along incompressible imbedded toriand Klein bottles into manifolds which have locally homogeneous Riemannianstructures. It is natural to ask whether the incompressilble tori and Klein bot-tles used in the torus decomposition can be taken to be minimal with respectto any Riemannian metric on M .

This question provides some motivation for the next existence theorem forminimal surfaces, due to Schoen and Yau [72] or Sacks and Uhlenbeck [69] withdifferent proofs. But the theory also applies to minimal tori in Riemannianmanifolds of arbitrary dimension.

For k ∈ 0 ∪ N, we let

Map(k)(T 2,M) = f ∈ C2(T 2,M) : the image off] : π1(T 2) −→ π1(M) is an abelian group with k generators .

For example, if f](π1(T 2)) ∼= Z ⊕ Z2, then f ∈ Map(2)(T 2,M). Note that themapping class group Γ = SL(2,Z) preserves Map(k)(T 2,M), so Eα induces amap

Eα :M(k)(T 2,M) −→ R, where M(k)(T 2,M) =Map(k)(T 2,M)× T

Γ.

Moreover, if f ∈ Map(2)(T 2,M),

f φ = f for some φ ∈ Γ ⇒ φ = identity,

so the mapping class group SL(2,Z) acts freely on Map(2)(T 2,M) × T . Thusif M(k)

α (T 2,M) denotes the completion with respect to the L2α1 norm, then

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M(k)α (T 2,M) will be a smooth Banach manifold. Recall that the α-energy

descends to a C2 map on the quotient

Eα :M(k)α (T 2,M) −→ R.

We expect that sequences tending to a minimum for α-energy in M(1)α (T 2,M)

would degenerate to a closed geodesic.

Lemma 3.11.1. The map Eα :M(2)α (T 2,M)→ R satisfies Condition C.

What Condition C asserts is that if [fi, ωi] is a sequence of points inM(2)(Σ,M)on which Eα is bounded and for which ‖dEα([fi, ωi])‖ → 0, and if for each i,(fi, ωi) ∈ Map(2)(T 2,M) × T is a representative for [fi, ωi], then there areelements φi ∈ Γ such that a subsequence of (fi φi, φ∗iωi) converges to a criticalpoint for Eα on Map(2)(T 2,M)× T .

To prove this, we recall that for the torus, the Teichmuller space T is theupper half plane, and after a change of basis we can arrange that an elementω ∈ T lies in the fundamental domain

D = u+ iv ∈ C : v > 0,−(1/2) ≤ u ≤ (1/2), u2 + v2 ≥ 1 (3.58)

for the action of the mapping class group Γ = SL(2,Z). The moduli spaceR is obtained from D by identifying points on the boundary. The complextorus corresponding to ω ∈ T can be regarded as the quotient of C by theabelian subgroup generated by d and ωd, where d is any positive real number, oralternatively, this torus is obtained from a fundamental parallelogram spannedby d and ωd by identifying opposite sides. The fundamental parallelogram ofarea one can be regarded as the image of the unit square (t1, t2) ∈ R2 : 0 ≤ti ≤ 1 under the linear transformation(

t1

t2

)7→(xy

)=

1√v

(1 u0 v

)(t1

t2

),

where z = x + iy is the usual complex coordinate on C. A straightforwardcalculation gives a formula for the usual energy

E(f, ω) =12

∫P

(∣∣∣∣∂f∂x∣∣∣∣2 +

∣∣∣∣∂f∂y∣∣∣∣2)dxdy

=12

∫P

(v

∣∣∣∣ ∂f∂t1∣∣∣∣2 +

1v

∣∣∣∣ ∂f∂t2 − u ∂f∂t1∣∣∣∣2)dt1dt2,

P denoting the image of the unit square. The only way that ω can approach theboundary of Teichmuller space while remaining in the fundamental domain Dis for v →∞. The rank two condition implies that the maps t1 7→ f(t1, b) mustbe homotopically nontrivial for each choice of t2 = b, and hence the length in

164

M of t1 7→ f(t1, b) is bounded below by a positive constant c. This implies that

E(f, ω) ≥ 12

∫ 1

0

∫ 1

0

v

∣∣∣∣ ∂f∂t1∣∣∣∣2 dt1dt2≥ v

2

∫ 1

0

(length of t1 7→ f(t1, b))2db ≥ c2v

2(3.59)

by the Cauchy-Schwarz inequality, and hence Eα(f, ω) (which is ≥ E(f, ω))must approach infinity.

Suppose now that [fi, ωi] is a sequence of points in M(2)(T 2,M) on whichEα is bounded and for which ‖dEα([fi, ωi])‖ → 0, and for each i, (fi, ωi) ∈Map(T 2,M)×T is a representative for [fi, ωi]. Then the projection [ωi] ∈ R ofωi ∈ T is bounded, and must therefore have a subsequence which converges toan element [ω∞] ∈ R. Hence there are elements φi ∈ Γ such that a subsequenceof φ∗iωi converges to an element ω∞ ∈ T . Then Eα,ω∞(fi φi) is boundedand ‖dEα,ω∞(fi φi)‖ → 0, so by Condition C for Eα,ω∞ , a subsequence of(fi φi, φ∗iωi) converges to a critical point for Eα on Map(T 2,M)× T . Thisestablishes Condition C for the function Eα.

Just as for a generic perturbation of Eα,ω, one can establish Morse inequalitiesfor a generic perturbation E′α of Eα. Thus we have a Morse-Witten complex forthis function, and we can use it to investigate critical points of E in the limitas α→ 1.

The following theorem was proven by Schoen and Yau [72] and Sacks andUhlenbeck [69]:

Theorem 3.11.2. Every component of M(2)(Σ,M) contains an element [f, ω]which minimizes the function

E :M(2)(T 2,M) −→ R

If (f, ω) is a representative, then f : T 2 → M is conformal and harmonic withrespect to ω, and hence a minimal surface.

Proof: Let C be a component of M(2)(T 2,M). Choose a decreasing sequenceαm → 1 and for each αm, a corresponding critical point

[fm, ωm] ∈M(2)(T 2,M) which minimizes Eαm on C.

We can assume that

Eαm([fm, ωm])→ infE(f, ω) : [f, ω] ∈ C. (3.60)

Since [ωm] must lie in a bounded region of the Riemann moduli space R1,we can arrange that [ωm] converges after passing to a subsequence, and henceafter choosing suitable representatives, arrange that ωm converges to an elementω ∈ T1. By Theorem 3.8.7, we can pass to a further subsequence so that eitherfm converges uniformly on compact subsets of T 2 to an ω-harmonic map

165

f : T 2 → M or nonconstant minimal two-spheres bubble off as α → 1. But inthe latter case, we can perform a surgery just like we described in the proof ofTheorem 3.10.1 and obtain a new parametrized torus fm : T 2 → M which liesin the same component C and has smaller energy

Eαm(fm, ωm) < Eαm(fm, ωm)− ε02,

contradicting (3.60). So no bubbling can occur, and we obtain a critical point(f, ω) for E which minimizes E within the component C.

Remark 3.11.3. The local regularity result of Osserman and Gulliver impliesthat the minimal tori found by Theorem 3.11.2 are immersed. Using a variant ofthe tower construction, it was proven by Freedman, Hass and Scott [24] that thetori and Klein bottles needed for the torus decomposition of a prime compactoriented three-manifold M can be taken to be minimal.

3.12 Higher genus minimal surfaces*

The existence theory for incompressible minimal tori presented in the previoussection can be extended to surfaces of higher genus. Suppose that Σ is a compactoriented compact surface of genus g ≥ 2. We set

Map′(Σ,M) = f ∈ C2(Σ,M) : f] : π1(Σ)→ π1(M) is injective ,

define M′(Σ,M) to be the quotient space

M′(Σ,M) =Map′(Σ,M)× T

Γ,

where Γ is the mapping class group, and let M′α(Σ,M) be the completion ofM′(Σ,M) with respect to the L2α

1 norm.

Lemma 3.12.1. If Σ is a compact oriented surface of genus g ≥ 2, then themap Eα :M′α(Σ,M)→ R satisfies Condition C.

The proof makes use to two ingredients, a collar theorem of Halpern and Keen,and the structure of the Bers compactification of the moduli space R of confor-mal structures on Σ.

We suppose that Σ is given the hyperbolic metric of Gaussian curvature−1 corresponding to the conformal structure ω ∈ T . The collar theorem is afundamental result of Riemann surface theory, and states that if γ is a closedgeodesic of this metric of length ≤ k1, where k1 is a positive constant, thenthere is a collar region C ⊆ Σ of fixed area k2 > 0 about γ. Indeed, we canarrange that γ lifts to the map

γ : [0, l]→ H2 defined by γ(t) = exp(t+

iπ

2

),

166

and the collar region is of the form C = π(C), where

C = reiθ ∈ H2 : 1 ≤ r < el, π − θ0 < θ < θ0, where cot θ0 =k2

2l. (3.61)

Lemma 3.12.2. Suppose that Σ has a closed geodesic γ whose length withrespect to the hyperbolic metric corresponding to ω is l ≤ k1, and that C is thecollar region about γ described above. If f : Σ → M is any smooth map, thenthe ω-energy of f |C is at least k4/l, for some positive constant k4.

We consider the lift f : C →M , which has energy density

e(f) =12|df |2 ≥ 1

2|(f γθ)′(t)|2

|γ′θ(t)|2, where γθ(t) = exp (t+ iθ) .

Straightforward calculation shows that

|γ′θ(t)|2 =1

sin2 θand |(f γθ)′(t)|2 = r2

∣∣∣∣∣∂f∂r∣∣∣∣∣2

,

so e(f) ≥ r2 sin2 θ

2

∣∣∣∣∣∂f∂r∣∣∣∣∣2

.

Thus we find that

E(f |C) =∫ π−θ0

θ0

∫ el

1

e(f)drdθ

r sin2 θ

≥∫ π−θ0

θ0

∫ el

1

r

2

∣∣∣∣∣∂f∂r∣∣∣∣∣2

drdθ =12

∫ π−θ0

θ0

∫ l

0

∣∣∣∣∣∂f∂t∣∣∣∣∣2

dtdθ,

where we have set r = et in the last integral. But by the Cauchy-Schwarzinequality,

L(f γθ)2 =∫ l

0

∣∣∣∣∣∂f∂t∣∣∣∣∣ dt ≤ l

∫ l

0

∣∣∣∣∣∂f∂t∣∣∣∣∣2

dt,

and since f γθ is a closed curve in M which is not homotopic to a constant,L(γθ) ≥ k3, for some positive constant, k3. Hence

E(f, ω) ≥ E(f |C) ≥ k23(π − 2θ0)

2l,

which yields the assertion of the lemma, since θ0 → 0 as l→ 0.

To finish the proof of Lemma 3.12.1, we suppose that [fm, ωm] is a sequenceof points in M′α(Σ,M) such that Eα([fm, ωm]) is bounded. We claim that thesequence [ωm] must stay in a compact region of the moduli space R. To

167

see this, we make use of the Bers compactification of the moduli space R, asdescribed in Appendix B of [37]. After passing to a subsequence, we can assumethat [ωm] converges to a point in the compactification. But points in thecompactification are Riemann surfaces with nodes and and as one approaches aRiemann surface with nodes from inside the moduli space, some homotopicallynontrivial loop must have its length go to zero, and then the energy will go toinfinity by Lemma 3.12.2. Thus [ωm] must remain bounded, and after passingto a subsequence we can assume that [ωm] → [ω∞] ∈ R. Now we use the factthat Eα,ω∞ satisfies Condition C to conclude that Eα :M′α(Σ,M)→ R satisfiesCondition C.

Thus we have a well-behaved Morse theory for suitable perturbations of theα-energy on M′α(Σ,M), just as in the case of the torus.

Theorem 3.12.3. If Σ is a compact oriented Riemann surface of genus g ≥ 2,then every component of the space M′(Σ,M) of incompressible maps from Σto M contains an element [f, ω] which minimizes the function

E :M′(Σ,M) −→ R

If (f, ω) is a representative, then f : Σ → M is conformal and harmonic withrespect to ω, and hence a minimal surface.

To prove this, we use essentially the same argument as we used in the proof ofTherem 3.11.2.

3.13 Complex form of second variation

In order to apply the apply the Morse theory of α-harmonic maps to derivegeometric consequences, it is useful to be able to estimate the Morse index ofa harmonic map. Recall that by Corollary 3.7.2, the second variation of energyat a harmonic map f : Σ→M is given by the formula

d2Eω(f)(V,W ) =∫

Σ

〈Lf (V ),W 〉dA,

where Lf is the Jacobi operator , defined by

Lf (V ) = − 1λ2

[D

∂x D∂x

+D

∂y D∂y

+K(V )]. (3.62)

Recall that we say that an element V ∈ TfM is a Jacobi field along f ifLf (V ) = 0.

Theorem 3.13.1. The Jacobi operator can be written in the following complexform:

Lf (V ) = − 4λ2

[D

∂z D∂z

+R

(·, ∂f∂z

)∂f

∂z

],

168

where the Riemann-Christoffel curvature tensor R has been extended to becomplex linear.

Straightforward calculations show that on the one hand,

4D

∂z D∂z

=D

∂x D∂x

+D

∂y D∂y

+√−1(D

∂x D∂y− D

∂y D∂x

)=

D

∂x D∂x

+D

∂y D∂y

+√−1R

(∂f

∂x,∂f

∂y

),

while on the other,

4R(·, ∂f∂z

)∂f

∂z= R

(·, ∂f∂x

)∂f

∂x+R

(·, ∂f∂y

)∂f

∂y

+√−1[R

(·, ∂f∂x

)∂f

∂y−R

(·, ∂f∂y

)∂f

∂x

]= R

(·, ∂f∂x

)∂f

∂x+R

(·, ∂f∂y

)∂f

∂y−√−1R

(∂f

∂x,∂f

∂y

),

the last step following from the Bianchi symmetry. Substitution into (3.62) nowyields the Theorem.

Corollary 3.13.2. The second variation of ω-energy is given by the formula:

d2Eω(f)(V, V ) = 4∫

Σ

[∣∣∣∣DV∂z∣∣∣∣2 −⟨R(V ∧ ∂f∂z

), V ∧ ∂f

∂z

⟩]dxdy,

for sections V of the complex vector bundle E, where the bar indicates complexconjugate and we use the notation 〈R(x ∧ y), v ∧ w〉 = 〈R(x, y)w, v〉.

Proof: Simply substitute into (3.62) and integrate by parts.

Corollary 3.13.3. A section V of the subbundle L of f∗TM⊗C, characterizedby the fact that ∂f/∂z is a locally defined section of L, is a Jacobi field if andonly if it is holomorphic.

Proof: If σ is a holomorphic section of L, then σ = g(∂f/∂z) where g is aholomorphic function. It follows that

D

∂z D∂z

(σ) = 0 and R

(σ,∂f

∂z

)= 0.

We leave the converse to the reader.

Remark 3.13.4. Note first that since Lf is a real operator, real and imaginaryparts of Jacobi fields are also Jacobi fields.

Remark 3.13.5. The dimension of the space of holomorphic sections O(L) ofthe line bundle L can often be estimated by the Riemann-Roch theorem from the

169

theory of Riemann surfaces. (One can apply the usual Riemann-Roch theoremto L

¯since L

¯has a canonical holomorphic structure by the Koszul-Malgrange

Theorem 3.1.3. Alternatively, one can apply the Atiyah-Singer Index Theoremdirectly to the operator Z 7→ (DZ/∂z)dz and use the fact that this operator hasthe same symbol as the ∂-operator, avoiding the proof of the Koszul-MalgrangeTheorem.) First consider the case in which Σ has genus zero. In this case, onerecalls that there is exactly one holomorphic line bundle Hk over Σ = S2 (upto isomorphism) whose Chern class satisfies 〈c1Hk), [Σ]〉 = k, and the spaceof holomorphic sections of this bundle has complex dimension k + 1 if k ≥ 0.The reason for the notation Hk is that the bundle of first Chern class k is thek-th tensor power of a bundle H called the hyperplane bundle over S2. ThusL = Hk, where k = c1(L)[S2].

According to (3.7), the line bundle L defined by the harmonic map f hasfirst Chern class given by the formula

〈c1(L), [Σ]〉 = 2 +∑νp : p is a branch point of f .

In particular, 〈c1(L), [Σ]〉 is always positive and L always has a space of holo-morphic section which has complex dimension at least three. A holomorphicsection of L can be identified with a meromorphic section σ of the holomorphictangent bundle TΣ with the property that its divisor (σ) satisfies

(σ) +∑νpp : p is a branch point of f ≥ 0.

Counting up the possibilities, we see that there is a holomorphic section of Lwhich has every possible principal part at every branch point of f .

We can do a similar analysis for line bundles over a torus T 2, except that thistime the line bundles of a given Chern class form a torus of dimension two, andif c1(L)[T 2] ≥ 1, the dimension of holomorphic sections of L will miss by onethe dimension that would be possible if every principal part were realized. Thusin the case of the torus, there is a section σ of L such that the correspondingsection of the tangent bundle to T 2 has arbitrary principal part at all but onebranch point.

3.14 Isotropic curvature

Just as the theory of geodesics uncovers relations between curvature and topol-ogy of Riemannian manifolds (through classical theorems with names such asSynge’s theorem, Myers’ theorem and the theorem of Hadamard and Cartanamong others), one might hope to apply the theory of minimal surfaces to cur-vature and topology. Pursuing this idea, however, leads to a different notion ofcurvature. Just as studying the stability of geodesics leads to sectional curva-ture, we find stability theory for minimal surfaces leads to the notion of isotropiccurvature.

170

Recall that in normal coordinates (x1, . . . , xn) centered at p on a Riemannianmanifold M , the Riemannian metric can be expressed by a Taylor series

gij = δij −13

∑k,l

Rikjl(p)xkxl + higher order terms,

where the Rikjl’s are components of the Riemann-Christoffel curvature tensor.The curvature operator is defined in terms of these components to be the linearmap

R : Λ2TpM −→ Λ2TpM

such that

R

(∂

∂xi

∣∣∣∣p

∧ ∂

∂xj

∣∣∣∣p

)=∑k,l

Rijkl(p)∂

∂xk

∣∣∣∣p

∧ ∂

∂xl

∣∣∣∣p

.

If z and w are linearly independent elements of TpM⊗C, the sectional curvatureof the complex two-plane σ spanned by z and w is

〈R(z ∧ w), z ∧ w〉〈z ∧ w, z ∧ w〉

,

where the bar denotes complex conjugation. The complex two-plane σ is saidto be isotropic if 〈z, z〉 = 〈w,w〉 = 〈z, w〉 = 0.

Definition. The Riemannian manifold M is said to have positive isotropiccurvature if K(σ) > 0, whenever σ is an isotropic complex two-plane.

To see why isotropic curvature is related to stability properties of minimal sur-faces, we first recall the second variation formula for a harmonic map f : S2 →M (Corollary 3.13.3):

d2Eω(f)(V, V ) = 4∫

Σ

[∣∣∣∣DV∂z∣∣∣∣2 −⟨R(V ∧ ∂f∂z

), V ∧ ∂f

∂z

⟩]dxdy. (3.63)

We need one further ingredient:

Grothendieck Theorem 3.14.1. Any holomorphic line bundle over the Rie-mann sphere S2 = CP 1 divides into a holomorphic direct sum of holomorphicline bundles.

This theorem, proven in [30], allows us to write E = f∗TM ⊗C as a direct sumof line bundles,

E = L1 ⊕ L2 ⊕ · · · ⊕ Ln, where c1(L1)[S2] ≥ · · · ≥ c1(Ln)[S2].

Since the Riemannian metric is invariant under the Levi-Civita connection, itextends to a holomorphic complex bilinear form

〈·, ·〉 : E×E −→ C,

171

and in particular, E is isomorphic to its dual. Thus the line bundles in theabove sequence can be arranged so that c1(Li) = −c1(Ln−i+1) for each i.

If V is a holomorphic section of one of the line bundles Li inthe direct sumdecomposition of E, then

〈V, V 〉 : S2 → C and⟨V,∂f

∂z

⟩: S2 → C

are holomorphic, and hence

〈V, V 〉 = (constant),⟨V,∂f

∂z

⟩= (constant).

In particular, if V is a holomorphic section of Li where Li has positive first Chernclass, or more generally if V is any holomorphic section such that 〈V, V 〉 = 0,then

〈V, V 〉 =⟨V,∂f

∂z

⟩=⟨∂f

∂z,∂f

∂z

⟩= 0,

andV and

∂f

∂zspan an isotropic two-plane

at every point where neither vanishes. If M has positive isotropic curvatures, ittherefore follows from the index formula (3.63) that d2Eω(f)(V, V ) < 0.

Let m be the number of line bundle summands Li of E with positive firstChern number,

c1(L1)[S2] ≥ · · · ≥ c1(Lm)[S2] ≥ 1, c1(Lm+1)[S2] ≤ 0

and let E0 be the direct sum of all of the line bundle summands with zero Chernclass; the dimension of the space O(E0) of holomorphic sections of E0 is n−2m.If V1, . . . , Vm are nonzero holomorophic sections of L1, . . . ,Lm respectively, then〈Vi,E0〉 = 0 for 1 ≤ i ≤ m. If W1, . . . ,Wl is a basis for O(E0), then 〈Wi,Wj〉is constant, and therefore there is a maximal isotropic holomorphic subbundleI0 ⊂ E0) which has rank ≥ (1/2)(rank of E0)-1).

If V is the space of holomorphic sections of L1 ⊕ · · ·Lm ⊕ I0, then dimV ≥(1/2)(dimM − 1) and

V ∈ V ⇒ d2Eω(f)(V, V ) < 0.

Thus we have proven a lemma which will be needed for the proof of the nexttheorem:

Lemma 3.14.2. Suppose that M is a Riemannian manifold which has positiveisotropic curvature. Then the Morse index of any harmonic two-sphere f : S2 →M is at least (1/2)(dimM − 1).

The following theorem [49] generalizes the earlier classical sphere theorem dueto Berger, Klingenberg and Toponogov:

172

Sphere Theorem 3.14.3. Suppose that M is a compact smooth simplyconnected Riemannian manifold of dimension at least four which has positiveisotropic curvature. Then M is homeomorphic to a sphere.

The idea behind the proof is to show that M is a homotopy sphere and applythe solutions to the generalized Poincare conjecture in dimensions greater thanfour to conclude that M is homeomorphic to a sphere.

By definition, a compact connected manifold M of dimension n is a homo-topy sphere if πq(M) = 0, for all integers q such that 0 < q < n. Thus if M isnot a homotopy sphere there must be an integer q with 0 < q < n such thatπq(M) 6= 0, and we choose the smallest such integer. By the Hurewicz isomor-phism theorem, q is also the smallest positive integer such that Hq(M ; Z) 6= 0.By Poincare duality, we can assume that q ≤ n/2.

Just as in §5.4, we let M be the space of smooth maps from S2 to Mand let π : M → M be the evaluations map defined by π(f) = f(0) where0 ∈ S2 = C ∪ ∞ is the basepoint. Since the fibration π possesses a section,the long exact homotopy sequence of π splits, and if Mp denotes the subset ofM consisting of maps such that f(0) = p, we conclude that

πk(M) ∼= πk(M)⊕ πk(Mp) ∼= πk(M)⊕ πk+2(M),

which implies thatπq−2(M) ∼= πq(M) 6= 0.

We now apply Morse theory to a perturbation Eα,ψ of the energy E, where ψis chosen so that all nonconstant critical points of Eα,ψ are Morse nondegenerate,and let α→ 1 and ψ → 0, in accordance with Theorem 2 from §3.5. We therebyobtain a sequence fm of critical points for Eαm,ψm , each having Morse indexno more than q−2. By the bubbling argument presented in §3.9, we find that asubsequence (still denoted by fm) converges uniformly in every Ck on everycompact subset of S2 − p1, · · · , pl, where p1, . . . , pl are a finite number ofbubble points, to a harmonic map on Σ− p1, · · · , pl, which can be extendedto a smooth harmonic map f∞ : Σ → M by the Sacks-Uhlenbeck removeablesingularity theorem.

Suppose first that f∞ is nonconstant. In that case, we claim that the Morseindex of f∞ is no larger than q − 2. For this, we need the following lemma:

Lemma 3.14.4. Suppose that fm : Σ→M is a sequence of αm-harmonic mapsin M which converge in Ck on compact subsets of Σ−x1, · · · , xl to a smoothharmonic map f∞ : Σ→M . Then

Morse index of f∞ ≤ lim inf Morse index of fm.

To prove the Lemma, we first recall the formula the second variation formula

173

(3.40) for α-harmonic two-spheres:

d2Eα(f)(V,W ) = α

∫Σ

(1 + |df |2)α−1[〈∇V,∇W 〉 − 〈K(V ),W 〉]dA

+ 2α(α− 1)∫

Σ

(1 + |df |2)α−2〈df,∇V 〉〈df,∇W 〉dA.

Note that the second term is dominated by the first and goes to zero when αis close to one, and the first term approaches d2E(f)(V,W ) when the supportof V and W does not contain any bubble points. We claim that if d2E(f∞) isnegative definite on a fixed linear space V of dimension q − 2, so is d2Eα(fm)when m is sufficiently large.

To see this, we make use of a cutoff function near the bubble points. First,define a map φ : R→ R by

φ(r) =

0, if r ≤ ε2,(log(ε2)− log r)/(log ε), if ε2 ≤ r ≤ ε,1, if ε ≤ r,

so that

dφ

dr(r) =

0, if r ≤ ε2,(−1)/(r log ε), if ε2 ≤ r ≤ ε,0, if ε ≤ r,

and ∫ 2π

0

∫ ε

0

(dφ

dr(r))2

rdrdθ =∫ ε

ε2

2πr(log ε)2

dr = − 2πlog ε

.

Thus if we define ψi : S2 → R so that it is one outside an ε-neightborhood ofthe bubble point xi and in terms of polar coordinates (ri, θi) about the bubblepoint xi satisfies the condition ψi = φ ri, then∫

S2|dψi|2dA ≤

−Clog ε

, where C is a positive constant,

and a similar estimate holds for ψ = ψ1 · · ·ψl, a cutoff function which vanishesat every bubble point.

If ε > 0 is chosen sufficiently small, then

V ∈ V ⇒ d2E(f)(ψV, ψV ) < 0 ⇒ d2Eα(f)(ψV, ψV ) < 0.

This shows that d2Eα(f) is negative definite on a space of dimension q− 2 andproves the Lemma.

It follows from Lemma 3.14.4 that if f∞ is nonconstant, it must have index≤ q − 2 ≤ (1/2)(dimM − 2) which contradicts Lemma 3.14.2.

But if f∞ is constant a nonconstant sphere must bubble off; that is theremust be a family of conformal reparametrizations gm : S2 → S2, such that

174

fm gm converges to a nonconstant harmonic sphere f∞ in Ck on compactsubsets of S2 − p1, · · · , pl, where p1, . . . , pl are a finite number of bubblepoints. We can choose pm ∈ S2 such that

‖dfm(pm)‖ = sup‖dfp‖ : p ∈ S2,

and after rotations we can arrange that all the pm’s are equal and in fact thatz(pm) = 0, where z is the standard coordinate on S2 − ∞ = C. Let rm =‖dfm(pm)‖, and let gm : S2 → S2 be the conformal map expressed in terms ofthe standard coordinate as hm(z) = rmz. Finally, let hm = fm gm, a sequenceof maps which converge to f∞ on compact subsets of S2 − p1, · · · , pl.

We must now replace d2Eα(fm) by d2Eα(fm gm) in the above argument.Once again, one obtains a nonconstant two-sphere f∞ in the limit. Moreover,a calculation similar to that for f∞ leads to the conclusion that f∞ has index≤ q − 2 ≤ (1/2)(dimM − 2), which once again contradicts Lemma 3.14.2.

Thus M must be a homotopy sphere, and follows from positive resolutionsof the generalized Poincare conjecture ([51] and [23]) in dimensions ≥ 4 that Mis homeomorphic to a sphere.

Remark 3.14.5. It can be shown that if the real sectional curvatures K(σ) ofa Riemannian manifold satisfy the inequalities

14< K(σ) ≤ 1, (3.64)

then the manifold has positive isotropic curvatures, but not conversely. Thesphere theorem proven by Berger, Klingenberg and Toponogov made the hy-pothesis (3.64) on real sectional curvatures and is therefore weaker than thesphere theorem we have proven. The complex projective space with the stan-dard Fubini-Study metric is simply connected, has real sectional curvatureslying in the range [1/4, 1] and has nonnegative isotropic curvature, but is nothomeomorphic to a sphere.

Remark 3.14.6. Conformally flat four-manifolds of positive scalar curvatureautomatically have positive isotropic curvature. It is possible to construct aconformally flat metric of positive scalar curvature on the connected sum ofa finite number of S3 × S1’s. Therefore the fundamental group of a compactmanifold of positive isotropic curvature can be a free group of arbitrary rank.However, a recent article of Fraser [22] shows that the fundamental group ofsuch a manifold cannot contain a free abelian group of rank two.

175

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