Alexandru Ioan Cuza University of Ia³isbarna/resurse/Nistor_TezaDoctorat.pdf · Alexandru Ioan...

Alexandru Ioan Cuza University of Ia³i

Faculty of Mathematics

PhD Thesis

BIHARMONICITY AND BICONSERVATIVITY TOPICS

IN THE THEORY OF SUBMANIFOLDS

Advisor, PhD Student,

Prof.Dr. Cezar ONICIUC Simona NISTOR (married BARNA)

IAI, 2017

Contents

Introduction v

1 Preliminaries 1

1.1 Some generalities on Riemannian manifolds . . . . . . . . . . . . . . . . 1

1.2 Some generalities on Riemannian submanifolds . . . . . . . . . . . . . . 7

1.3 CMC surfaces in 3-dimensional space forms . . . . . . . . . . . . . . . . 11

1.4 The energy and bienergy functionals . . . . . . . . . . . . . . . . . . . . 14

1.5 Biharmonic and biconservative submanifolds . . . . . . . . . . . . . . . . 17

2 Biconservative surfaces in 3-dimensional space forms 27

2.1 Biconservativity and minimality in N3(c) . . . . . . . . . . . . . . . . . 27

2.2 An intrinsic characterization of biconservative surfaces in N3(c) . . . . . 35

3 Complete biconservative surfaces in R3 and S3 49

3.1 Complete biconservative surfaces in R3 . . . . . . . . . . . . . . . . . . . 49

3.1.1 Uniqueness of complete biconservative surfaces in R3 . . . . . . . 59

3.2 Complete biconservative surfaces in S3 . . . . . . . . . . . . . . . . . . . 68

4 Biconservative surfaces in arbitrary Riemannian manifolds 107

4.1 More characterizations of biconservative submanifolds . . . . . . . . . . 107

4.2 Properties of biconservative surfaces . . . . . . . . . . . . . . . . . . . . 111

4.3 A Simons type formula for S2 . . . . . . . . . . . . . . . . . . . . . . . . 119

4.3.1 Exemples of submanifolds with ∇AH = 0 . . . . . . . . . . . . . 124

Bibliography 125

iii

Introduction

The introduction is, rstly, concerned with presenting some of the ideas that encouraged

the study of the geometry of biharmonic and biconservative submanifolds and, secondly,

with briey describing the new results that we have obtained and are presented in the

next chapters.

In the last few years the theory of biconservative submanifolds proved to be a very

interesting research topic (see, for example, [15,28,30,31,49,6770]). This theory arose

from the theory of biharmonic submanifolds, but the class of biconservative submanifolds

is richer than the later one. For this reason, we have been focused on the study of

biconservative submanifolds.

Let (Mm, g) and (Nn, h) be two Riemannian manifolds. A biharmonic map, as sug-

gested by J. Eells and J.H. Sampson in [26], is a critical point of the bienergy functional

E2 : C∞(M,N) → R, E2(φ) =

1

2

∫M

|τ(φ)|2 vg,

where τ(φ) is the tension eld of a smooth map φ : M → N , with respect to the xed

metrics g and h. The corresponding Euler-Lagrange equation, obtained by G.Y. Jiang

in [38], is

τ2(φ) = −∆φτ(φ)− traceg RN (dφ, τ(φ))dφ = 0,

where τ2(φ) is the bitension eld of φ, ∆φ = − traceg(∇φ∇φ −∇φ

∇)is the rough

Laplacian dened on sections of φ−1(TN) and RN is the curvature tensor eld of N

given by

RN (X,Y )Z = [∇NX ,∇N

Y ]Z −∇N[X,Y ]Z.

An isometric immersion φ : (Mm, g) → (Nn, h) or, simply, a submanifold M of N , is

called biharmonic if φ is a biharmonic map. Any harmonic map is, clearly, biharmonic.

Therefore, we are interested in studying biharmonic and non-harmonic maps which

are called proper-biharmonic. As a submanifold M of N is minimal if and only if

v

vi Chapter 0. Introduction

φ : (M, g) → (N,h) is a harmonic map, by a proper-biharmonic submanifold we mean

a non-minimal biharmonic submanifold.

Following D. Hilbert ([34]), to an arbitrary functional E we can associate a symmet-

ric tensor eld S of type (1, 1), called the stress-energy tensor, which is conservative, i.e.,

divS = 0, at the critical points of E. In the particular case of the bienergy functional

E2, G.Y. Jiang ([39]) dened the stress-energy tensor S2, also called the stress-bienergy

tensor, by

⟨S2(X), Y ⟩ =1

2|τ(φ)|2⟨X,Y ⟩+ ⟨dφ,∇τ(φ)⟩⟨X,Y ⟩

− ⟨dφ(X),∇Y τ(φ)⟩ − ⟨dφ(Y ),∇Xτ(φ)⟩,

and proved that

divS2 = −⟨τ2(φ), dφ⟩.

Therefore, if φ is biharmonic, i.e., is a critical point of E2, then divS2 = 0. The

variational meaning of S2 was given in [42].

One can see that if φ : (Mm, g) → (Nn, h) is an isometric immersion, then divS2 =

0 if and only if the tangent part of the bitension eld associated to φ vanishes. A

submanifold M is called biconservative if divS2 = 0.

The biconservative submanifolds were studied for the rst time in 1995 by Th.

Hasanis and Th. Vlachos in [33]. In that paper the biconservative hypersurfaces in the

Euclidean space Rn were called H-hypersurfaces, and they were fully classied in R3 and

R4. Actually, the authors were looking for biharmonic hypersurfaces in R4, and their

strategy was to determine rst the hypersurfaces with τ2(φ)⊤ = 0 and then to check if

such hypersurfaces also satisfy τ2(φ)⊥ = 0. They proved that none of the hypersurfaces

with τ2(φ)⊤ = 0 satises τ2(φ)

⊥ = 0, except the minimal ones. We mention that we

prefer to use the term biconservative submanifolds, instead of H-submanifolds as

they are characterized by the vanishing of the divergence of S2 (the authors in [33] did

not use the stress-bienergy tensor).

On the other hand, the study of submanifolds with constant mean curvature, i.e.,

CMC submanifolds, and of minimal submanifolds, has been a very active research topic

in Dierential Geometry for more than 50 years. Recent monographs on these topics

are, for example, [41, 47,48]. There are two ways to develop new research directions:

• to study CMC submanifolds which satisfy some additional geometric hypotheses

(for example, CMC and biharmonicity);

• to study hypersurfaces in space forms, i.e., hypersurfaces in spaces with constant

sectional curvature, which are highly non-CMC.

The study of biconservative surfaces matches with both directions from above.

vii

Indeed, biconservative submanifolds in arbitrary manifolds (and in particular, bicon-

servative surfaces) which are also CMC have some remarkable properties, as we will

see in Chapter 4.

The CMC hypersurfaces in space forms are trivially biconservative, so more in-

teresting is the study of biconservative hypersurfaces which are non-CMC, i.e., with

grad f = 0 at any point of an open subset. Moreover, where grad f = 0, a biconser-

vative surface in a 3-dimensional space form, i.e., a 3-dimensional space with constant

sectional curvature, satises

f∆f + | grad f |2 + 4

3cf2 − f4 = 0,

where ∆ is the Laplace-Beltrami operator on M , i.e., the mean curvature satises a

second order PDE.

We would also want to underline the fact that under the hypothesis of biconservativ-

ity some known results in the theory of submanifolds can be extended to more general

contexts. For example, the generalized Hopf function associated to a CMC biconservat-

ive surface in a Riemannian manifold is holomorphic (compare with the classical results:

the Hopf function associated to a CMC surface in a 3-dimensional space form N3(c)

is holomorphic, and the generalized Hopf function associated to a PMC surface in an

n-dimensional space form is holomorphic). Other example is that a pseudoumbilical

biconservative submanifold φ : Mm → Nn, with m = 4, is CMC, for any ambient

manifold N .

The thesis is organized as follows. In Chapter 1, we establish the notations and

recall some known results. It is divided in ve sections. In the rst section, we recall

properties of symmetric tensor elds of type (1, 1) on a Riemannian manifold, as their

rough Laplacian and their divergence. Then, we give some results about complete

metrics on manifolds and we end this section with few formulas and denitions from

conformal geometry and theory of distributions. In the second section, we present some

basic facts on Riemannian submanifolds: the Gauss, Codazzi and Ricci equations (i.e.,

the basic equations of submanifolds) and the fundamental theorem for submanifolds.

Further, we specialize in the next section on CMC surfaces in 3-dimensional space

forms. We recall here the famous Ricci problem:

Given an abstract surface, which are the necessary and sucient conditions such that it

admits a minimal immersion in N3(c)?

We note that minimal surfaces in spheres, of higher codimension, which satisfy the

Ricci condition were studied, for example, in [73], where the author proved the Lawson's

conjecture (which state that if(M2, g

)is a non-at abstract surface satisfying the Ricci

condition and φ :(M2, g

)→ Sn is a minimal immersion, then φ has a certain form) in

some particular cases.

viii Chapter 0. Introduction

In the fourth section, we present the energy and bienergy functionals E and E2, and

the associated stress-energy tensor S and the stress-bienergy tensor S2. We also give

here the variational meaning of these tensor elds. In the last section, we present some

characterizations formulas for biharmonic and biconservative submanifolds. Then, we

focus on biconservative surfaces in N3(c) with grad f = 0 at any point and recall some

known properties of them.

The main results in this chapter are Theorem 1.42 and Theorem 1.45, obtained

in [15], which describe the properties of biconservative surfaces in 3-dimensional space

forms with grad f = 0 at any point, and Theorem 1.49, obtained in [27], which is a

uniqueness result and says that:

If we have an abstract surface which admits two biconservative immersions in N3(c)

such that the gradients of their mean curvature functions are dierent from zero at any

point, then these two immersions dier by an isometry of N3(c).

Although the major part of the results in this chapter is known, we also present here

few original results as Corollary 1.39, Theorem 1.40 (these two results are presented here

for the rst time), and Theorem 1.49 (this already appeared in [27]).

Chapter 2 contains two sections. We note that the explicit local equations of

biconservative surfaces in a 3-dimensional space form N3(c), with grad f = 0 at any

point, were obtained in [15] and [30]. Moreover, in [15], it is shown that the Gaussian

curvature of a such biconservative surface in a 3-dimensional space form satises the

following equation

(c−K)∆K − | gradK|2 − 8

3K(c−K)2 = 0, (0.1)

that is very similar with that used by G. Ricci-Curbastro [64] in 1895 to characterize,

intrinsically, minimal surfaces in R3.

As we will see in the rst section, we can use this property of biconservative surfaces

to prove results similar to those in [40], [53], or [64], in our context. More precisely,

given an abstract surface which satises intrinsic equation (0.1) with c = 0, by a simple

conformal transformation, it becomes a Ricci surface in R3 (see Theorem 2.4). The

case when c = 0 is dierent. Given an abstract surface which admits a biconservative

immersion in N3(c) with grad f = 0 at any point (thus, a stronger condition than (0.1)),

there exists a conformal transformation of the metric on the surface (this time more

complicated), such that the surface with the new metric is a Ricci surface in N3(c) (see

Theorem 2.7).

An implication of the fact that an abstract surface(M2, g

)admits a biconservative

immersion in N3(c), with grad f = 0 at any point, is that the level curves of K are

ix

circles in M with constant curvature

κ =3| gradK|8(c−K)

, (0.2)

a condition found in [15]. We begin the second section with Theorem 2.10 which gives

some equivalent conditions to (0.2).

Another important result in this chapter is Theorem 2.17 which state the following:

If an abstract surface satisfying c−K > 0, gradK = 0 at any point and (0.2), admits

a biconservative immersion in N3(c), then grad f = 0 (and it is unique).

One of the main results in this thesis is Theorem 2.18 which says that:

An abstract surface admits locally a biconservative embedding in N3(c), with grad f = 0

at any point, if and only if c−K > 0, gradK = 0 at any point and (0.2) holds.

Therefore, even if the notion of biconservative submanifolds belongs, obviously, to

the extrinsic geometry, in the particular case of biconservative surfaces in N3(c), they

admit also an intrinsic characterization.

The chapter ends with Theorem 2.22 that gives other three equivalent conditions

to (0.2). This time, these conditions are expressed in terms of certain isothermal co-

ordinates. This result will be very helpful in the construction process of complete

biconservative surfaces in R3 and in S3, as we will see in the next chapter.

We note that most of the results from this chapter are original and they can be also

found in [27] and [56]. Some results, as Theorem 2.10, Theorem 2.17, Theorem 2.21,

appear here for the rst time.

The goal of Chapter 3 is to construct complete biconservative surfaces in R3 and

S3. We start with the local extrinsic and intrinsic results and extend them to the

global extrinsic and intrinsic results. The local extrinsic problem consists in nding

all biconservative surfaces in 3-dimensional space forms with grad f = 0 at any point

and the local intrinsic problem is to determine all abstract surfaces(M2, g

)satisfying

c−K > 0, gradK = 0 at any point of M , and condition (0.2).

We have seen that the local extrinsic and intrinsic problems are completely solved,

and they are equivalent since given an abstract surface(M2, g

), it admits a biconser-

vative immersion in N3(c) with grad f = 0 at any point of M if and only if c−K > 0,

gradK = 0 at any point ofM and (0.2) holds. Moreover, this immersion is unique, and

thus we have a bijection between the set of biconservative immersions in N3(c), with

grad f = 0 at any point, where c ∈ R is xed, and the set of abstract surfaces which

satisfy the above conditions.

Then, we consider the global problem, again, from extrinsic and intrinsic point of

view. To solve the global extrinsic problem means to determine all complete biconser-

vative surfaces in 3-dimensional space forms which satisfy grad f = 0 at any point of

x Chapter 0. Introduction

an open and dense subset. To solve the global intrinsic problem means to determine

all complete abstract surfaces(M2, g

)which on that open and dense subset satisfy

c−K > 0, gradK = 0 and the level curves of K are the circles in M with curvature κ

given in (0.2).

We cannot say that the global extrinsic and intrinsic are equivalent. As we will see,

the global extrinsic problem implies the global intrinsic problem, but, even if M would

be simply connected, the converse implication we do not know to be true, because we

cannot dene a tensor eld A on the whole M .

The two global problems are not completely solved, in the sense that we construct

examples of complete biconservative surfaces in N3(c), c = 0 and c = 1, but without

proving their uniqueness.

As we said, for the global extrinsic problem, we have asked grad f = 0 at any point

of an open and dense subset. We impose this hypothesis because we belive that

Conjecture 1. Let M2 be a biconservative surface in N3(c). If there exists an open

subset U of M such that grad f = 0 on U , then grad f = 0 on M .

In Theorem 3.16 and in Theorem 3.21 we prove Conjecture 1 in some particular

cases. We suppose that the proof of Conjecture 1 will follow from the analysis of the

PDE obtained in Corollary 1.39. We also believe that

Conjecture 2. The only complete simply connected biconservative surfaces in R3 or S3,with grad f = 0 at any point of an open and dense subset, are those given in Theorem

3.11 and Theorem 3.48, respectively.

In Theorem 3.17 and in Theorem 3.22 we prove Conjecture 2 in some particular

cases.

Summarising the above two conjectures, we can state the following

Conjecture 3. The only complete simply connected non-CMC biconservative surfaces

in R3 or S3 are those given by Theorem 3.11 and Theorem 3.48, respectively.

Moreover, we can also state the following open problem that would follow from the

quality of the biconservative immersion given in Theorem 3.48 of being double periodic

or not.

Open problem. Does there exist a non-CMC biconservative surface in S3 that is

compact?

This chapter has two sections. In the rst section, we consider the global problem

and construct complete biconservative surfaces in R3, with grad f = 0 at any point

of an open dense subset. We determine such surfaces in two ways. One way is to

use the local extrinsic characterization of biconservative surfaces in R3 and to glue

xi

two pieces together in order to obtain a complete biconservative surface (Theorem 3.5).

The other way is more analytic and consists in using the local intrinsic characterization

theorem in order to obtain a complete biconservative immersion from(R2, gC0

)in R3

with grad f = 0 on an open dense subset of R2; here, C0 is a positive constant and

therefore we obtain a one-parameter family of solutions (Theorem 3.11). It is worth

mentioning that, by a simple transformation of the metric gC0 ,(R2,

√−KC0gC0

)is

(intrinsically) isometric to a helicoid (Theorem 3.12).

The rst section ends with a subsection, where we study the uniqueness of complete

biconservative surfaces in R3. One of the main results in this chapter is Theorem 3.20,

which state that:

If S is a compact biconservative regular surface in R3, then S is CMC and thus a round

sphere.

Moreover, we prove a stronger theorem, Theorem 3.22, which is one of the most

important result of this thesis, and says that:

If S is a complete non-CMC biconservative regular surface in R3, then S = SC0, where

SC0is the complete biconservative surface of revolution in R3 given in Theorem 3.5.

In the second section, we consider the global problem of biconservative surfaces in

S3, with grad f = 0 at any point of an open dense subset. As in the R3 case, we use

the local extrinsic classication of biconservative surfaces in S3, but now the gluing

process is not as clear as in R3. Further, we change the point of view and use the

local intrinsic characterization of biconservative surfaces in S3. We construct complete

Riemannian surfaces(R2, gC1,C∗

1

)which admit a biconservative immersion in S3 with

grad f = 0 on an open dense subset of R2 and we show that, up to isometries, there

exists only a one-parameter family of such Riemannian surfaces indexed by C1 (Theorem

3.48). The above construction consists in two steps: rst, we obtain a complete surface

of revolution in R3, whose universal cover is(R2, gC1,C∗

1

), and second, we determine

explicitly the biconservative immersion in S3. Theorem 3.48 is one of the main results

in this thesis.

We note that most of the results from this chapter are original and they were presen-

ted also in [54], [56], and [57]. Moreover, the results in Subsection 3.1.1 are presented

here for the rst time.

In the last chapter, Chapter 4, we use dierent techniques as in the previous

chapters, and study in a uniform manner the properties of biconservative surfaces in

arbitrary Riemannian manifolds.

The chapter is divided in three sections. In the rst section, we present some charac-

terizations of biconservative submanifolds which satisfy additional geometric hypotheses

xii Chapter 0. Introduction

as AH being a Codazzi tensor eld (Proposition 4.7), or the submanifold being PMC,

i.e., having the mean curvature vector eld H parallel in the normal bundle (Proposi-

tion 4.11). We also study the properties of submanifolds with AH parallel, as they are

automatically biconservative (Proposition 4.1).

In the second section, we focus on biconservative surfaces. As biconservative surfaces

are characterized by divS2 = 0, where S2 is a symmetric tensor eld of type (1, 1), some

of their properties will follow from general properties of a symmetric tensor eld of type

(1, 1) with free-divergence as they are presented in Theorem 4.18. In Theorem 4.19, we

nd the link between biconservativity, the property of the shape operator AH to be a

Codazzi tensor eld, the holomorphicity of a generalized Hopf function and the quality

of the surface to have constant mean curvature.

Another result in this section is Theorem 4.26, which gives a description of the

metric and of the shape operator AH for a CMC biconservative surface in an arbitrary

manifold. This description is done in terms of |H| and the principal curvatures of M ,

i.e., the eigenvalues functions of AH .

We end this section by proving that a biconservative surface in an arbitrary manifold,

with constant principal curvatures, can be immersed in N3(c) having either AH or S2,

as shape operator (Theorem 4.31 and Theorem 4.32).

In the last section, we nd the expression of the rough Laplacian∆RS2 of S2 and then

we determine a Simons type formula (Proposition 4.34). A consequence of Proposition

4.34 is Theorem 4.39, which states:

If φ :M2 → Nn is a compact CMC biconservative surface and K ≥ 0, then ∇AH = 0

and M is at or pseudoumbilical.

With a dierent technique we get a similar result in the complete non-compact case

(Theorem 4.41).

Almost all of the results from this chapter are original and they can be also found

in [55]. Some of them are known results, but obtained in a dierent way.

Acknowledgements. Firstly, I would like to express my sincere gratitude to my

advisor Prof.Dr. Cezar Oniciuc for his useful ideas, his patience, motivation and constant

support throughout this thesis. His invaluable guidance helped me in all the time of

research and writing of this thesis.

Beside my advisor, I would like to thank the others members of my PhD committee:

Prof.Dr. Ioan Buc taru, Prof.Dr. Dorel Fetcu and Prof.Dr. R zvan Liµcanu, for their

insightful comments and encouragement.

I am also grateful to the Doctoral School of Faculty of Mathematics, for the nancial

support, and to the Faculty of Mathematics, Alexandru Ioan Cuza University of Ia³i,

xiii

for hospitality and for the opportunity to have access to its infrastructure. My sincere

thanks go to the following Financing Programs and their directors, that supported me

in this research:

• Constant mean curvature and biharmonic submanifolds, PN-II-RU-TE-2014-4-

0004; Director: Prof.Dr. Dorel Fetcu.

• Variational methods with applications to generalized vector optimization problems,

PN-II-RU-TE-2014-4-0019; Director: Prof.Dr. Marius Durea.

• The European Social Fund through Sectoral Operational Programme Human

Resources Development 2007 2013, Towards a New Generation of Elite Re-

searchers through Doctoral Scolarships, POSDRU/187/1.5/S/155397; Director:

Assoc.Prof. Liviu-George Maha.

Last but not the least, I would like to thank my family: my husband, my parents

and my brother, for supporting me spiritually throughout writing this thesis, for their

love and constant motivation.

Chapter 1Preliminaries

In this chapter, we explain the notations and recall some basic results which will be

used throughout the thesis.

We will not dene the fundamental notions of dierentiable and Riemannian geo-

metry, as dierentiable manifolds, vector bundles, linear connections, Riemannian man-

ifolds, etc., as they are supposed to be known.

Conventions. Throughout this thesis, all manifolds, metrics and maps are assumed

to be smooth, i.e. of class C∞, and we will often indicate the various Riemannian

metrics by the same symbol ⟨·, ·⟩. All manifolds and submanifolds are assumed to be

connected and oriented. Also, by a CMC submanifold we understand a submanifold

with constant mean curvature dierent from zero, and a non-CMC hypersurface is a

hypersurface with grad f = 0 at any point of an open subset W of M , where f denotes

the mean curvature function on M , and W is not necessarily the whole M , i.e., M \Wcan be a non-empty set.

Most of the results in this chapter are known and we will mainly follow the mono-

graphs [11, 12, 16, 17, 22, 60, 63], to present them. Though, we note that the results in

Corollary 1.39 and in Theorem 1.40 appear here for the rst time, and the result given

in Theorem 1.49 is also original, but was presented for the rst time in [27].

1.1 Some generalities on Riemannian manifolds

It is well-known that a symmetric tensor eld T of type (1, 1) on a Riemannian manifold

(Mm, g) can be identied with a symmetric tensor eld T of type (0, 2) by

⟨T (X), Y ⟩ = T (X,Y ), X, Y ∈ C(TM),

1

2 Chapter 1. Preliminaries

and, henceforth, we will use the same notation T instead of T .

Denition 1.1. Let (Mm, g) be a Riemannian manifold and ∇ its Levi-Civita connec-

tion. Consider the vector elds on M , X, Z, Yj , j = 1, r, the tensor eld T of type

(1, r), the symmetric tensor eld of type (0, 2), S, and Xii=1,m a local orthonormal

frame eld on M . Then

(i) the divergence of a vector eld is a smooth function given by

divX = trace (Z → ∇ZX)

=

m∑i=1

⟨∇XiX,Xi⟩;

(ii) the divergence of a tensor eld of type (1, r) is a tensor eld of type (0, r) dened

by

(div T ) (Y1, Y2, · · · , Yr) = trace (Z → (∇ZT ) (Y1, Y2, · · · , Yr))

=m∑i=1

⟨(∇XiT ) (Y1, Y2, · · · , Yr) , Xi⟩;

(iii) the divergence of a symmetric tensor eld of type (0, 2) is a tensor eld of type

(0, 1) given by

(divS)(X) =m∑i=1

(∇S) (Xi, Xi, X) =m∑i=1

(∇XiS) (Xi, X) .

Moreover, if T is a symmetric tensor eld of type (1, 1), then

(div T )(X) =

m∑i=1

⟨(∇XiT ) (X), Xi⟩ =m∑i=1

⟨(∇XiT ) (Xi) , X⟩ = ⟨trace(∇T ), X⟩,

i.e., div T = (trace(∇T ))♯ or, equivalently, (div T ) = trace(∇T ), where ♯ and are themusical isomorphisms.

Denition 1.2. Let (Mm, g) be a Riemannian manifold and α be a smooth function

on M . Then

(i) the gradient of a smooth function is a vector eld dened by

gradα =m∑i=1

(Xiα)Xi,

where Xii=1,m is a local orthonormal frame eld on M ;

1.1. Some generalities on Riemannian manifolds 3

(ii) the Hessian of a smooth function is a symmetric tensor eld of type (1, 1) given

by

(Hessα)(X) = ∇X gradα, X ∈ C(TM).

Moreover, we have

⟨(Hessα)(X), Y ⟩ = ⟨(Hessα)(Y ), X⟩ = X(Y α)− (∇XY )α, X, Y ∈ C(TM),

and therefore, we can think the Hessian as a symmetric tensor eld of type (0, 2).

Clearly,

∆α = − trace(Hessα).

Proposition 1.3. Let (Mm, g) be a Riemannian manifold and consider T and S two

symmetric tensor elds of type (1, 1). Then

⟨∆RT, S⟩ = ⟨∇T,∇S⟩ − divZ, (1.1)

with ∆RT = − trace∇2T , Z ∈ C(TM), Z = ⟨∇XiT, S⟩Xi, where Xii=1,m is an

orthonormal local frame eld.

Proposition 1.4. Let (Mm, g) be a Riemannian manifold and consider T a symmetric

tensor eld of type (1, 1) and α a smooth function on M . Then

div (T (gradα)) = ⟨div T, gradα⟩+ ⟨T,Hessα⟩, (1.2)

Proposition 1.5 (The Ricci formula). Let (Mm, g) be a Riemannian manifold and T

a tensor eld of type (1, 1) (not necessarily symmetric). Then(∇2T

)(X,Y, Z)−

(∇2T

)(Y,X,Z) = R(X,Y )T (Z)− T (R(X,Y )Z),

where X,Y, Z ∈ C(TM).

Denition 1.6. A symmetric tensor eld T of type (1, 1) is called a Codazzi tensor

eld if

(∇T )(X,Y ) = (∇T )(Y,X), X, Y ∈ C(TM).

Remark 1.7. If ∇T = 0, then T is a Codazzi tensor eld.

Further, we present some results about the completeness of Riemannian metrics.

Proposition 1.8 ([32]). Let Mm be a manifold and consider g and g two metrics on

it. If (M, g) is complete and g − g is non-negative denite at any point of M , i.e.,

(g − g) (X,X) ≥ 0, for any X ∈ C(TM), then (M, g) is also complete.


Even if this result is known, we did not nd a proof of it, and this is the reason why

we will present now our proof.

Proof. Let d and d be the distance functions determined by g and g, respectively. Since

(M, g) is complete, (M,d) is a complete metric space, i.e., every Cauchy sequence of

points in M converges to a point in M , with respect to the distance d. If we denote

g1 = g − g, clearly g1(X,X) ≥ 0, for any X ∈ C(TM) and g = g1 + g.

Consider p and q two points in M and γ : [a, b] → M a piece-wise smooth curve

such that γ(a) = p and γ(b) = q. Let a = t0 < t1 < · · · < tn = b be a partition of the

interval [a, b] such that γ[ti,ti+1] is smooth, for any i = 0, n− 1. The length of γ, with

respect to the metric g, is given by

l(γ; g) =n−1∑i=0

∫ ti+1

ti

√g (γ′(τ), γ′(τ)) dτ,

and the length of the same γ, with respect to the metric g, is given by

l (γ; g) =

n−1∑i=0

∫ ti+1

ti

√g (γ′(τ), γ′(τ)) dτ.

Obviously,

l (γ; g) =

n−1∑i=0

∫ ti+1

ti

√g (γ′(τ), γ′(τ)) + g1 (γ′(τ), γ′(τ)) dτ

≥n−1∑i=0

∫ ti+1

ti

√g (γ′(τ), γ′(τ)) dτ

= l(γ; g),

and, then, one obtains

d(p, q) ≤ d(p, q). (1.3)

In order to prove that(M, d

)is a complete metric space, we consider a Cauchy sequence,

(pn)n∈N∗ ⊂ M , with respect to d, i.e., for any ε > 0, there exists nε ∈ N∗, such that

for any m,n ∈ N∗ with m,n > nε, we have d (pm, pn) < ε. From (1.3) it follows that

(pn)n∈N∗ is Cauchy with respect to d. Therefore, since (M,d) is a complete metric space,

there exists p0 ∈M such that (pn)n∈N∗ converges to p0, with respect to the metric d.

On the other side, the topology of the submanifold M coincide with the topology

induced by the metric space (M,d). Thus, d and d determine the same topology on M .

Let ε0 > 0 arbitrary xed. Denote by

Bd (p0; ε0) =q ∈M | d (q, p0) < ε0

.

1.1. Some generalities on Riemannian manifolds 5

Since Bd (p0; ε0) is open and p0 ∈ Bd (p0; ε0), there exists r = rε0 > 0 such that

Bd (p0; r) ⊂ Bd (p0; ε0) .

Now, since (pn)n∈N∗ converges to p0, with respect to the metric d, for r, there exists

nr ∈ N∗ such that for any n > nr we have

pn ∈ Bd (p0; r) ⊂ Bd (p0; ε0) .

Therefore, (pn)n∈N∗ converges to p0, with respect to the metric d, i.e.,(M, d

)is a

complete metric space.

Further, we present a similar result with the above one.

Proposition 1.9. Let Mm be a manifold and consider g and g two metrics on it.

If (M, g) is complete and there exists a > 0 such that g(X,X) ≥ ag(X,X), for any

X ∈ C(TM), then (M, g) is also complete.

Proof. The proof of this result is, in fact, the same as that of Proposition 1.8, with only

one dierence. Here,

l (γ; g) =

n−1∑i=0

∫ ti+1

ti

√g (γ′(τ), γ′(τ)) dτ

≥ an−1∑i=0

∫ ti+1

ti

√g (γ′(τ), γ′(τ)) dτ

= a l(γ; g),

and, therefore

d(p, q) ≤ 1

ad(p, q).

In fact, Proposition 1.9 also appear in [16], where is state the following problem.

Proposition 1.10. Let S and S two regular surfaces and φ : S → S a dieomorphism.

Assume that S is complete and there exists a > 0 such that

Ip (w) ≥ aIφ(p) (dφ(p) (w)) , w ∈ TpS, (1.4)

for any p ∈ S, where I and I denote the rst fundamental form of S and S, respectively.

Then S is complete.


Proof. We consider i : S → R3 and i : S → R3 the canonical inclusions of S and S in

R3 and ⟨·, ·⟩0 the Euclidean metric on R3. Denote by g = i∗⟨, ⟩0 and g = φ∗ (i∗⟨, ⟩0) themetrics induced by R3 on S, and by φ on S, respectively.

With these notations, our problem can be reformulated as: given a surface S with

g and g two metrics on it, if (S, g) is complete (because S is complete) and

g(X,X) ≥ ag(X,X), X ∈ C(TM),

then (S, g) is complete. This is obviously true, from Proposition 1.9.

From the conformal geometry, we recall the following known formulas.

Let(M2, g

)be a Riemannian surface with the Levi-Civita connection ∇, curvature

tensor eld R, Gaussian curvature K and Laplacian ∆. Consider a new Riemannian

metric g = e2ρg on M , where ρ ∈ C∞(M). Then

• the Levi-Civita connection, with respect to g, is given by

∇XY = ∇XY + (Xρ)Y + (Y ρ)X − g(X,Y ) grad ρ, X, Y ∈ C(TM); (1.5)

• the curvature tensor eld, with respect to g, is equal to

R(X,Y )Z = R(X,Y )Z − (∇dρ)(Y, Z)X + (∇dρ)(X,Z)Y + (Y ρ)(Zρ)X−

− (Xρ)(Zρ)Y −∇g(Y,Z)X−g(X,Z)Y grad ρ−

− | grad ρ|2 (g(Y, Z)X − g(X,Z)Y ) + (Xρ)g(Y, Z) grad ρ−

− (Y ρ)g(X,Z) grad ρ, X, Y, Z ∈ C(TM);

• the Riemann-Christoel tensor eld, with respect to g, is given by

R(X,Y, Z,W ) = e2ρ(R(X,Y, Z,W )− (∇dρ)(Y,W )g(X,Z)+

+ (∇dρ)(X,W )g(Y, Z)− (∇dρ)(X,Z)g(Y,W )+

+ (∇dρ)(Y, Z)g(X,W )− (Xρ)(Wρ)g(Y, Z)+

+ (Y ρ)(Wρ)g(X,Z) + (Xρ)(Zρ)g(Y,W )−

− (Y ρ)(Zρ)g(X,W )− | grad ρ|2g(Y,W )g(X,Z)+

+ | grad ρ|2g(X,W )g(Y, Z)), X, Y, Z,W ∈ C(TM);

• the Gaussian curvature, with respect to g, is determined by

K = e−2ρ(K +∆ρ); (1.6)

1.2. Some generalities on Riemannian submanifolds 7

• the Laplacian, with respect to g, can be computed by

∆α = e−2ρ∆α, α ∈ C∞(M); (1.7)

We end this section with some results concerning the distributions.

Let Nn be a manifold of dimension n = m+ k. Assume that to each point p ∈ N is

assigned anm-dimensional subspace Dp of TpN and in a neighborhood U of each p ∈ N ,

there are m vector elds Xi ∈ C(TU), i = 1,m, such that Xi(p)i=1,m ⊂ TpN is a

basis of Dq, for any q ∈ U . Then, we say that D is a m-plane distribution of dimension

m on N and Xii=1,m is a local frame eld of D. Moreover, the distribution D is called

involutive if there exists a local frame eld Xii=1,m around any point such that

[Xi, Xj ] = αkijXk, i, j ∈ 1,m,

where αkij are local smooth functions.

We note that the above notion does not depend of the chosen local frame eld.

Now, if D is a distribution on N and M is a submanifold of N such that for any

q ∈M , TqM ⊂ Dq, then M is called integral manifold of D.We have to notice that an integral manifold may be of lower dimension than D, and

is not necessary a regular manifold.

Finally, a distribution D onNn of dimensionm, where n = m+k, is called completely

integrable if for each point p ∈ N there exists a local chart (U ;φ) =(U ;x1, · · · , xn

)with φ(U) = Cnε (0), ε > 0, and ∂xii=1,m is a local frame eld on U for D, where∂xi = ∂

∂xiand

∂∂xα

α∈1,n is the natural frame eld corresponding to the local chart

(U ;φ). Note that, in this case, there exists an m-dimensional integral manifold M

through each point q ∈ U such that TqM = Dq, i.e. dimM = m. Indeed, an m-slice

dened by xm+1 = constant, · · · , xn = constant, is an m-dimensional integral manifold

of D. Of course, a completely integrable distribution is involutive since

[∂xi , ∂xj ] = 0, i, j ∈ 1,m.

In fact, we have the following result.

Theorem 1.11 (Frobenius). A distribution D on a manifold N is completely integrable

if and only if it is involutive.

1.2 Some generalities on Riemannian submanifolds

A submanifold of a given Riemannian manifold (Nn, h) is a pair (Mm, φ), where Mm

is a manifold and φ : M → N is an immersion. We always consider on M the induced


metric g = φ∗h, thus φ : (M, g) → (N,h) is an isometric immersion (for simplicity we

write φ :M → N without mentioning the metrics). We also write φ :M → N , or even

M , instead of (M,φ).

In order to x the notations, we recall the rst order fundamental equations of a sub-

manifold in a Riemannian manifold, as these equations dene the second fundamental

form, the shape operator and the connection in the normal bundle. Let φ :Mm → Nn

be an isometric immersion. For each p ∈ M , Tφ(p)N can be written as the orthogonal

direct sum

Tφ(p)N = dφ(TpM)⊕ dφ(TpM)⊥, (1.8)

and NM =∪p∈M

dφ(TpM)⊥ is referred to as the normal bundle of φ (or of M), in N .

Denote by ∇ and ∇N the Levi-Civita connections on M and N , respectively, and

by ∇φ the induced connection in the pull-back bundle φ−1(TN) =∪p∈M

Tφ(p)N . Taking

into account the decomposition in (1.8), one has the Gauss formula

∇φXdφ(Y ) = dφ(∇XY ) +B(X,Y ), X, Y ∈ C(TM),

where B ∈ C(⊙2T ∗M ⊗NM) is called the second fundamental form of M in N . Here

T ∗M denotes the cotangent bundle of M . The mean curvature vector eld of M in N

is dened by H = (traceB)/m ∈ C(NM), where the trace is considered with respect

to the metric g.

Furthermore, if η ∈ C(NM), then one has the Weingarten formula

∇φXη = −dφ(Aη(X)) +∇⊥

Xη, X ∈ C(TM),

where Aη ∈ C(T ∗M ⊗ TM) is called the shape operator of M in N in the direction

η, and ∇⊥ is the induced connection in the normal bundle. Moreover, ⟨B(X,Y ), η⟩ =⟨Aη(X), Y ⟩, for all X,Y ∈ C(TM), η ∈ C(NM). In the case of hypersurfaces, we

denote f = traceA, where A = Aη and η is the unit normal vector eld, and we have

H = (f/m)η; f is the (m times) mean curvature function.

A submanifold M of N is called a PMC submanifold if H is parallel in the normal

bundle and dierent from zero, and a CMC submanifold if |H| is constant and dierent

from zero.

When confusion is unlikely we locally identify M with its image by φ, X with

dφ(X) and ∇φXdφ(Y ) with ∇N

XY . With this in mind, we can write the Gauss and the

Weingarten formulas as

∇NXY = ∇XY +B(X,Y ),

and

∇NXη = −Aη(X) +∇⊥

Xη.

1.2. Some generalities on Riemannian submanifolds 9

Proposition 1.12 (The Gauss equation). Let φ :Mm → Nn be a submanifold. Then

⟨RN (X,Y )Z,W ⟩ = ⟨R(X,Y )Z,W ⟩ − ⟨B(X,W ), B(Y, Z)⟩+ ⟨B(Y,W ), B(X,Z)⟩,

where X,Y, Z,W ∈ C(TM).

If Mm is a hypersurface in Nm+1(c), then the Gauss equation becomes

R(X,Y )Z = c (⟨Y, Z⟩X − ⟨X,Z⟩Y ) + ⟨A(Y ), Z⟩A(X)− ⟨A(X), Z⟩A(Y )

for any X,Y, Z ∈ C(TM).

Moreover, if M2 is a surface in N3(c), we can rewrite the Gauss equation as

K = detA+ c,

where K is the Gaussian curvature of the surface.

Proposition 1.13 (The Codazzi equation). Let φ :Mm → Nn be a submanifold. Then

(∇XAη) (Y )− (∇YAη) (X) = A∇⊥Xη

(Y )−A∇⊥Y η

(X)−(RN (X,Y )η

)⊤,

where X,Y ∈ C(TM) and η ∈ C(NM), or, equivalently,(∇⊥XB)(Y, Z)−

(∇⊥YB)(X,Z) =

(RN (X,Y )Z

)⊥, X, Y, Z ∈ C(TM).

In particular, if N has constant sectional curvature, i.e., N = Nn(c), where c ∈ R, theCodazzi equation becomes

(∇XAη) (Y )− (∇YAη) (X) = A∇⊥Xη

(Y )−A∇⊥Y η

(X), X, Y ∈ C(TM), η ∈ C(NM),

or, equivalently,(∇⊥XB)(Y, Z) =

(∇⊥YB)(X,Z), X, Y, Z ∈ C(TM).

Moreover, if Mm is a hypersurface in Nm+1(c), then the Codazzi equation becomes

(∇XA) (Y )− (∇YA) (X) = 0.

Using the Codazzi equation, we easily nd the next result.

Proposition 1.14. Let φ :Mm → Nn be a submanifold. Then

trace∇AH =m

2grad

(|H|2

)+ traceA∇⊥

· H(·) + trace

(RN (·,H)·

)⊤. (1.9)

Corollary 1.15. Let φ :Mm → Nn(c) be a submanifold, c ∈ R. Then

trace∇AH =m

2grad

(|H|2

)+ traceA∇⊥

· H(·). (1.10)


Proposition 1.16 (The Ricci equation). Let φ :Mm → Nn be a submanifold. Then(RN (X,Y )η

)⊥= R⊥(X,Y )η +B (Aη(X), Y )−B (X,Aη(Y )) ,

where X,Y ∈ C(TM) and η ∈ C(NM), or, equivalently,

⟨RN (X,Y )η, ξ⟩ = ⟨R⊥(X,Y )η, ξ⟩ − ⟨[Aη, Aξ]X,Y ⟩,

where X,Y ∈ C(TM), η, ξ ∈ C(NM), and [Aη, Aξ] = AηAξ −AξAη.

Denition 1.17. A hypersurface φ : Mm → Nm+1 is called umbilical if A = (f/m)I,

where I is the identity tensor eld of type (1, 1).

Denition 1.18. A submanifold φ : Mm → Nn is called pseudoumbilical if AH =

|H|2I, where I is the identity tensor eld of type (1, 1).

Theorem 1.19. Let φ :Mm → Nm+1(c) an umbilical hypersurface. Then M is CMC.

Next, we recall the fundamental theorem for submanifolds.

Theorem 1.20. (The fundamental theorem for submanifolds)

(i) Let Mm be a simply connected Riemannian manifold, π : E → M a Riemannian

vector bundle of rank p with a compatible connection ∇, and let B be a symmetric

section of the homeomorphism bundle Hom(TM ×TM,E) ≡ (TM ⊗TM)∗⊗E =

(TM)∗ ⊗ (TM)∗ ⊗ E, i.e., B : C(TM)× C(TM) → E is a C∞(M) bilinear and

symmetric map. Dene, for each local section η of E, a map Aη : C(TM) →C(TM) by

⟨Aη(X), Y ⟩ = ⟨B(X,Y ), η⟩, X, Y ∈ C(TM).

If B and ∇ satisfy the Gauss, Codazzi and Ricci equations for the case of con-

stant sectional curvature c, then there is an isometric immersion φ : Mm →Nn=m+p(c), and a vector bundle isomorphism φ : C(E) → C(NM) along φ, such

that for every X,Y ∈ C(TM) and any η, ξ local sections of E:

⟨φ(η), φ(ξ)⟩ = ⟨η, ξ⟩, φ(B(X,Y )) = B(X,Y ), φ(∇Xη) = ∇⊥X φ(η),

where B and ∇⊥ are the second fundamental form, and the normal connection of

φ, respectively.

(ii) Suppose that φ and ψ are two isometric immersions of a connected manifold Mm

into Nn=m+k(c). Let NMφ, Bφ and ∇⊥φ denote the normal bundle, the second

fundamental form and the normal connection of φ, respectively; and let NMψ,

Bψ and ∇⊥ψ be the corresponding objects for ψ. If there exists a vector bundle

1.3. CMC surfaces in 3-dimensional space forms 11

isomorphism ˜φ : C(NM)φ → C(NM)ψ such that, for every X,Y ∈ C(TM) and

every η, ξ ∈ C(NM)φ:

⟨ ˜φ(η), φ(ξ)⟩ = ⟨η, ξ⟩, ˜φ(Bφ(X,Y )) = Bψ(X,Y ), ˜φ(∇⊥φXη) = ∇⊥

ψX

˜φ(η),

then there is an isometry F : Nn(c) → Nn(c) such that

ψ = F φ and dF |NMφ= ˜φ.

Theorem 1.21. (The fundamental theorem for hypersurfaces)

(i) Let Mm be a simply connected Riemannian manifold and let A : C(TM) →C(TM) be a symmetric tensor eld of type (1, 1) satisfying the Gauss and Codazzi

equations in the case of constant sectional curvature c. Then there is an isometric

immersion φ : Mm → Nm+1(c) such that A = Aη, for some unit normal vector

eld η ∈ C(NM), where Aη denotes the shape operator of the immersion φ.

(ii) Let φ :Mm → Nm+1(c) and ψ :Mm → Nm+1(c) be two connected hypersurfaces,

and let φ : C(NM)φ → C(NM)ψ be one of the two vector bundle isomorphisms.

Suppose that Bψ(X,Y ) = ˜φ(Bφ(X,Y )) or Bψ(X,Y ) = − ˜φ(Bφ(X,Y )), for every

X,Y ∈ C(TM), where Bψ and Bφ denote, respectively, the second fundamental

forms of ψ and φ. Then there exists an isometry F : Nm+1(c) → Nm+1(c) such

that ψ = F φdF |NMφ

= ˜φ or dF |NMφ= − ˜φ

.

1.3 CMC surfaces in 3-dimensional space forms

We recall now, a classical result concerning the existence of CMC surfaces in three-

dimensional space forms, i.e., in 3-dimensional spaces with constant sectional curvature

N3(c).

Theorem 1.22. ([40]) Let φ :(M2, g

)→ N3(c) be a CMC surface. Then |H|2 + c−

K ≥ 0 at any point, and either |H|2 + c −K = 0 everywhere, i.e., M is umbilical, or

|H|2 + c−K = 0 only at isolated points. Moreover, on the set where |H|2 + c−K > 0,

we have

∆log(|H|2 + c−K

)+ 4K = 0, (1.11)

or, equivalently,(|H|2 + c−K

)∆K − | gradK|2 + 4K

(|H|2 + c−K

)2= 0. (1.12)


Proof. Let φ :(M2, g

)→ N3(c) be a CMC surface and η ∈ C(NM), |η| = 1. Consider

b ∈ C(⊙2T ∗M

)such that

b(X,Y ) = ⟨B(X,Y ), η⟩ = ⟨A(X), Y ⟩, X, Y ∈ C(TM).

Let (U ;u, v) an isothermal chart onM such that g = e2ρ(du2 + dv2

), where ρ = ρ(u, v)

is a smooth function on U . If we denote by

b11 = b (∂u, ∂u) , b12 = b (∂u, ∂v) , b22 = b (∂v, ∂v) ,

the metric g can be rewritten as

g = b11du2 + 2b12dudv + b22dv

2.

The matrix of the shape operator A with respect to ∂u, ∂v is

A = e−2ρ

(b11 b12

b12 b22

).

Since A is a symmetric tensor eld of type (1, 1), traceA = f and f2 = 4|H|2, it followsthat we can consider a smooth function h1 on U such that

b11 = h1 +f

2e2ρ and b22 = −h1 +

f

2e2ρ.

As usual, we denote ∂z = (∂u − i∂v) /2. Then

b (∂z, ∂z) =1

4(b11 − b22 − 2ib12) ,

and substituting the expressions of b11 and b22 from above, we nd

2b (∂z, ∂z) = h1 − ih2,

where h2 = b12.

The matrix of A with respect to ∂u, ∂v becomes

A = e−2ρ

h1 +f2 e

2ρ h2

h2 −h1 + f2 e

2ρ

.

From the Gauss equation, detA = K − c, we get |h|2 = e4ρ(|H|2 + c−K

), where

h = h1 − ih2. It follows that |H|2 + c−K ≥ 0 at any point of U .

By a straightforward computation, it can be seen that the Codazzi equation is

equivalent to h = h1 − ih2 being holomorphic.

1.3. CMC surfaces in 3-dimensional space forms 13

Since the zeros of a non-zero holomorphic function are isolated, we have |H|2 +

c −K = 0 only at such isolated points. Since U was arbitrary chosen, it follows that

|H|2 + c−K ≥ 0 at any point of M and the zeros of |H|2 + c−K are isolated points.

On the set where |H|2+c−K > 0, as h is holomorphic, it follows that ∆log |h|2 = 0

and then

∆log(|H|2 + c−K

)+ 4K = 0.

By a straightforward computation, we obtain

∆log(|H|2 + c−K

)= − 1

|H|2 + c−K

(∆K − 1

|H|2 + c−K| gradK|

),

and using this relation, it is easy to see that (1.11) is equivalent to (1.12).

Remark 1.23. ([64]) If φ :(M2, g

)→ N3(c) is a minimal surface, i.e., H = 0, the

conclusions of Theorem 1.22 also hold.

Remark 1.24. If φ :(M2, g

)→ N3(c) is a CMC surface with constant Gaussian

curvature, then either M is umbilical or M is a at isoparametric surface with no

umbilical points. The isoparametric surfaces in N3(c) are well-known.

We also have a (kind of) converse result of Theorem 1.22.

Theorem 1.25. ([40]) Let(M2, g

)be an abstract surface, c ∈ R and a ∈ R∗

+. Assume

that on M we have a2 + c−K > 0 and

4K +∆ log(a2 + c−K

)= 0.

Then(M2, g

)admits locally a CMC embedding in N3(c) with |H| = a.

Proof. Let (U ;u, v) an isothermal chart on M such that g = e2ρ(du2 + dv2

), where

ρ = ρ(u, v) is a smooth function on U . Assume that U is simply connected (and also

connected). From (1.6), it is easy to see that K = ∆ρ = ∆ log(e4ρ)/4. Substituting K

in

4K +∆ log(a2 + c−K

)= 0,

we obtain ∆log((a2 + c−K

)e4ρ)= 0. Thus, there exists a holomorphic function h

such that

|h|2 =(a2 + c−K

)e4ρ. (1.13)

The function h is unique up to a multiplicative complex number of type eiθ. We consider

h = h1−ih2 and dene a tensor eld A of type (1, 1) on U by A (∂u) =(e−2ρh1 + a

)∂u+

e−2ρh2∂v and A (∂v) = e−2ρh2∂u +(−e−2ρh1 + a

)∂v. Then A, can be written, with

respect to ∂u, ∂v, as

A = e−2ρ

(h1 + e2ρa h2

h2 −h1 + e2ρa

).


Using (1.13), it is easy to prove that A satises the Gauss equation detA = K − c.

By a straightforward computation, since h is holomorphic, it follows that A satises

the Codazzi equation (∇∂uA) (∂v) = (∇∂vA) (∂u). Thus, there exists an isometric im-

mersion from U in N3(c) having A as its shape operator. Since traceA = 2a and

| traceA| = 2|H|, we get that |H| = a is a constant, i.e, the immersion is CMC.

If fact, as h is determined up to a multiplicative complex number of norm 1, we

have a one-parameter family of CMC immersions, all with |H| = a. We can restrict

the domain U such that these immersions become embeddings.

Remark 1.26. ([64]) If a = 0 in Theorem 1.25, then,(M2, g

)admits locally a minimal

embedding in N3(c).

1.4 The energy and bienergy functionals

A harmonic map φ : (Mm, g) → (Nn, h) between two Riemannian manifolds is a critical

point of the energy functional

E : C∞(M,N) → R, E(φ) =1

2

∫M

|dφ|2 vg,

and it is characterized by the vanishing of its tension eld

τ(φ) = traceg∇dφ.

For general accounts on the theory of harmonic maps we could see, for example, the

monographs [8, 23,25,71] and the papers [24,26].

The idea of the stress-energy tensor associated to a functional comes from D. Hilbert

([34]). Thus, to a given functional E, one can associate a symmetric 2-covariant tensor

eld S such that divS = 0 at the critical points of E. When E is the energy functional,

P. Baird and J. Eells in [4], and A. Sanini in [66], dened the tensor eld

S = e(φ)g − φ∗h =1

2|dφ|2g − φ∗h,

and proved that

divS = −⟨τ(φ), dφ⟩.

Hence, S can be chosen as the stress-energy tensor of the energy functional. It is

worth mentioning that S also has a variational meaning. Indeed, we can x a map

φ :Mm → (Nn, h) and think E as being dened on the set of all Riemannian metrics on

M . The critical points of this new functional are these Riemannian metrics determined

by the vanishing of their stress-energy tensor S.

1.4. The energy and bienergy functionals 15

More precisely, we assume that M is compact and denote by

G = g : g is a Riemannian metric on M .

For a deformation gt of g we consider ω = ddt

∣∣t=0

gt ∈ TgG = C(⊙2T ∗M

)and dene

a new functional

F : G → R, F(g) = E(φ).

Thus, we have the following result.

Theorem 1.27 ([4,66]). Let φ :Mm → (Nn, h) and assume that M is compact. Then

d

dt

∣∣∣∣t=0

F (gt) =1

2

∫M⟨ω, e(φ)g − φ∗h⟩ vg.

Therefore g is a critical point of F if and only if its stress-energy tensor S vanishes.

We mention here that, if φ : (Mm, g) → (Nn, h) is an arbitrary isometric immersion,

then divS = 0.

A natural generalization of harmonic maps is represented by biharmonic maps. A

biharmonic map φ : (Mm, g) → (Nn, h) between two Riemannian manifolds is a critical

point of the bienergy functional

E2 : C∞(M,N) → R, E2(φ) =

1

2

∫M

|τ(φ)|2 vg,

and is characterized by the vanishing of its bitension eld

τ2(φ) = −∆φτ(φ)− traceg RN (dφ, τ(φ))dφ,

where

∆φ = − traceg(∇φ∇φ −∇φ

∇)

is the rough Laplacian of φ−1TN and the curvature tensor eld is

RN (X,Y )Z = ∇NX∇N

Y Z −∇NY ∇N

XZ −∇N[X,Y ]Z, X, Y, Z ∈ C(TM).

We note that the biharmonic equation τ2(φ) = 0 is a fourth-order non-linear elliptic

equation and that any harmonic map is biharmonic. A non-harmonic biharmonic map

is called proper-biharmonic.

The theory of biharmonic maps, which represents the most natural generalization

of biharmonic functions, is an old subject with the origin in the theory of elasticity and

uid mechanics (see, for example, [1,46]). Nowadays, this theory is well developed and

we can see, for example, [2, 3, 57,9, 29,37,44,45,58,61,72].


In [39], G.Y. Jiang dened the stress-energy tensor S2 of the bienergy (also called

the stress-bienergy tensor) by

⟨S2(X), Y ⟩ =1

2|τ(φ)|2⟨X,Y ⟩+ ⟨dφ,∇τ(φ)⟩⟨X,Y ⟩

− ⟨dφ(X),∇Y τ(φ)⟩ − ⟨dφ(Y ),∇Xτ(φ)⟩,

as it satises

divS2 = −⟨τ2(φ), dφ⟩.

As in the harmonic case, the tensor eld S2 has a variational meaning too. We x

a map φ :Mm → (Nn, h) and dene a new functional

F2 : G → R, F2(g) = E2(φ).

Then we have the following result.

Theorem 1.28 ([42]). Let φ :Mm → (Nn, h) and assume that M is compact. Then

d

dt

∣∣∣∣t=0

F2 (gt) = −1

2

∫M⟨ω, S2⟩ vg,

so g is a critical point of F2 if and only if S2 = 0.

We mention that, if φ : (Mm, g) → (Nn, h) is an isometric immersion, then divS2

does not necessarily vanish.

We end this section with the following properties of the stress-bienergy tensor.

Proposition 1.29. Consider a submanifold φ :Mm → Nn. Then we have:

(i) the stress-bienergy tensor of φ is determined by

S2 = −m2

2|H|2I + 2mAH ; (1.14)

(ii) traceS2 = m2|H|2(2− m

2

);

(iii) the relation between the divergence of S2 and the divergence of AH is given by

divS2 = −m2

2grad

(|H|2

)+ 2m divAH ; (1.15)

(iv) |S2|2 = m4|H|4(m4 − 2

)+ 4m2 |AH |2.

Remark 1.30. From equation (1.15), we see that if divS2 = 0 it does not follow

that divAH automatically vanishes. In fact, only when |H| is constant divS2 = 0 is

equivalent to divAH = 0.

1.5. Biharmonic and biconservative submanifolds 17

1.5 Biharmonic and biconservative submanifolds

A submanifold φ : Mm → Nn is called biharmonic if the isometric immersion φ is a

biharmonic map from (Mm, g) to (Nn, h).

Even if the notion of biharmonicity may be more appropriate for maps than for

submanifolds, as the domain and the codomain metrics are xed and the variation is

made only through maps, the biharmonic submanifolds proved to be interesting objects

(see, for example, [61]).

If divS2 = 0 for a submanifold M in N , then M is called biconservative. Thus, M

is biconservative if and only if the tangent part of its bitension eld vanishes.

We have the following characterization theorem of biharmonic submanifolds, ob-

tained by splitting the bitension eld in the tangent and normal part.

Theorem 1.31. A submanifold Mm of a Riemannian manifold Nn is biharmonic if

and only if

traceA∇⊥· H

(·) + trace∇AH + trace(RN (·,H)·

)⊤= 0

and

∆⊥H + traceB (·, AH(·)) + trace(RN (·,H)·

)⊥= 0,

where ∆⊥ is the Laplacian in the normal bundle.

Various forms of the above result were obtained in [18,42,59]. From here we deduce

some characterization formulas for the biconservativity.

Proposition 1.32. Let φ :Mm → Nn be a submanifold. Then the following conditions

are equivalent:

(i) M is biconservative;

(ii) traceA∇⊥· H

(·) + trace∇AH + trace(RN (·,H)·

)⊤= 0;

(iii) m2 grad

(|H|2

)+ 2 traceA∇⊥

· H(·) + 2 trace

(RN (·,H)·

)⊤= 0;

(iv) 2 trace∇AH − m2 grad

(|H|2

)= 0.

The following properties are immediate.

Proposition 1.33. LetMm be a submanifold of a Riemannian manifold Nn. If ∇AH =

0, then M is biconservative.

Proposition 1.34. Let Mm be a submanifold of a Riemannian manifold Nn. Assume

that N is a space form and M is PMC. Then M is biconservative.


Proposition 1.35 ([10]). Let Mm be a pseudoumbilical submanifold of a Riemannian

manifold Nn with m = 4. Then M is a CMC submanifold.

If we consider the particular case of hypersurfaces, Theorem 1.31 gives the following

result

Theorem 1.36 ([10, 62]). If Mm is a hypersurface in a Riemannian manifold Nm+1,

then M is biharmonic if and only if

2A(grad f) + f grad f − 2f(RicciN (η)

)⊤= 0,

and

∆f + f |A|2 − f RicciN (η, η) = 0,

where η is the unit normal vector eld of M in N and f is the mean curvature function

of M .

Proposition 1.37. A hypersurface Mm in a space form Nm+1(c) is biconservative if

and only if

A(grad f) = −f2grad f. (1.16)

Corollary 1.38. Any CMC hypersurface in Nm+1(c) is biconservative.

Therefore, biconservative hypersurfaces may be regarded as the next natural topic

to be studied after CMC surfaces.

Considering the divergence in equation (1.16), using the fact that divA = grad f

and equation (1.2) with T = A and α = f , we obtain the following corollary.

Corollary 1.39. Let Mm be a biconservative hypersurface in Nm+1(c). Then

f∆f − 3| grad f |2 − 2⟨A,Hess f⟩ = 0.

Next, we show that the two distributions determined by grad f , where f is the mean

curvature function of a biconservative hypersurface in a space form, are completely

integrable. As a one-dimensional distribution is always integrable, we only must prove

the following result.

Theorem 1.40. Let Mm be a biconservative hypersurface in Nm+1(c) and assume that

grad f = 0 at any point of M . Then, the distribution D orthogonal to that determined

by grad f is completely integrable. Moreover, any integral manifold of D, of maximal

dimension, has at normal connection as a submanifold in Nm+1(c).


Proof. Since grad f = 0 at any point of M , there exists the global unit vector eld

X1 = grad f/| grad f |. Since M is a biconservative hypersurface in a space form with

grad f = 0 at any point of M , using (1.16), we easily get A (X1) = −(f/2)X1.

Let Xii=1,m be a local orthonormal frame eld on M . We note that Xkk=2,m is

a local basis of the distribution D orthogonal to that determined by grad f . It is clear

that grad f = (X1f)X1 and Xkf = 0, for any k ∈ 2,m.

As grad f = 0, we can assume that there exists(U ;x1, · · · , xm

)a local chart on M

such that f = x1. We have

grad f = gij∂f

∂xi∂xj = g1j∂xj ,

and then one obtains ⟨grad f, ∂xk⟩ = 0, for any k ∈ 2,m. It follows that ∂x2 , · · · , ∂xmis a local basis for the distribution D and therefore D is completely integrable.

Further, let us denote by P an integral manifold of D, of maximal dimension, i.e. of

dimension m − 1, and by the indices 1 and 2 the objects corresponding to the normal

bundle of P in M , and in N , respectively. Consider the global unit normal vector eld

η of M in N . Clearly X1, η is a unit frame eld in the normal bundle of P in N .

Let Z ∈ C(TP ). It is clear that Z is also a tangent vector eld on M , and using

the Gauss formula (for M in N), one obtains

∇NZX1 = ∇M

Z X1 +B (Z,X1)

= ∇MZ X1 + ⟨B (Z,X1) , η⟩η

= ∇MZ X1 + ⟨A (X1) , Z⟩η,

where B is the second fundamental form of M in N and A is the shape operator of M

in N .

We note that A (Xi) = −(f/2)X1 is a normal vector eld of P in N , and then

∇NZX1 = ∇M

Z X1 ∈ C(TM).

Using the Weingarten formula (for P in N) one also has

∇NZX1 = −A2

X1(Z) +2 ∇⊥

ZX1.

Therefore, as A2X1

(Z) ∈ C(TP ), and then A2X1

(Z) ∈ C(TM), one gets 2∇⊥ZX1 ∈

C(TM). Since 2∇⊥Z , X1 is also a normal vector eld of P in N , it follows that 2∇⊥

ZX1

is collinear with X1. But |X1| = 1, and then⟨2∇⊥

ZX1, X1

⟩= 0. Finally, we obtain

2∇⊥ZX1 = 0. (1.17)

Similarly, we can see that 2∇⊥Zη ∈ C(TM). Indeed, as |η| = 1, we have ∇⊥

Zη = 0

and, from the Weingarten formula (forM in N), one obtains ∇NZ η = −A(Z) ∈ C(TM).


Now, using the Weingarten formula, but this time, for P in N , it follows that

∇NZ η = −A2

η(Z) +2 ∇⊥

Zη.

As A2η(Z) ∈ C(TM), we come to the conclusion.

Since 2∇⊥Zη ∈ C(TM) and is also a normal vector eld of P in N , one gets that

2∇⊥Zη is collinear with X1. Moreover, as ⟨X1, η⟩ = 0 and 2∇⊥

ZX1 = 0, we obtain

2∇⊥Zη = 0. (1.18)

Using (1.17) and (1.18), we conclude with 2R⊥ (Z1, Z2)σ = 0, for any Z1, Z2 ∈ C(TP )

and σ a normal vector eld of P in N .

Remark 1.41. The above result, which holds for biconservative hypersurfaces in any

space form Nm+1(c), extends the similar result for biconservative hypersurfaces in

Rm+1, obtained in [33].

Next, we study properties of biconservative surfaces in 3-dimensional space forms.

Theorem 1.42 ([15]). Let φ :M2 → N3(c) be a biconservative surface with grad f = 0

at any point of M . Then we have f > 0 and

f∆f + | grad f |2 + 4

3cf2 − f4 = 0, (1.19)

where ∆ is the Laplace-Beltrami operator on M .

Proof. Since grad f = 0 at any point ofM , we can consider X1 = (grad f)/| grad f | andX2 two vector elds such that X1(p), X2(p) is a positively oriented orthonormal basis

at any point p ∈M . In particular, we obtain that M is parallelizable. Using (1.16), we

have A (X1) = −(f/2)X1. Since traceA = f , we get A (X2) = 3f/2. Thus, the matrix

of A with respect to the (global) orthonormal frame eld X1, X2 is

A =

−f2 0

0 3f2

.

From the Gauss equation, K = c+ detA, we obtain

f2 =4

3(c−K). (1.20)

Thus c−K ≥ 0 on M .

Now, from the denitions of X1 and X2, we nd that

grad f = (X1f)X1 and X2f = 0.


Further, we consider the connection forms ωji on M dened by ∇Xi = ωjiXj , where

i, j ∈ 1, 2. Obviously, ωji = −ωij . Using the Codazzi equation

∇X1A (X2)−∇X2A (X1) = A [X1, X2] ,

and X2f = 0, we get

4fω12 (X1)X1 +

(3 (X1f)− 4fω1

2 (X2))X2 = 0.

If we assume that there exists a point p0 ∈ M such that f (p0) = 0, from the above

equation, one obtains (X1f) (p0) = 0. As X2f = 0, it follows that (grad f) (p0) = 0,

which is a contradiction. Therefore, f = 0 at any point of M , and we can assume that

f > 0. This leads to c−K > 0 on M,

ω12 (X1) = 0 and ω1

2 (X2) =3X1f

4f, (1.21)

so the Levi-Civita connection on M is given by

∇X1X1 = ∇X1X2 = 0, ∇X2X1 = −3X1f

4fX2, ∇X2X2 =

3X1f

4fX1. (1.22)

By a simple computation, it can be proved that the intrinsic expression for the Gaussian

curvature K of M is

K = X1

(ω12 (X2)

)−(ω12 (X2)

)2.

If we substitute ω12 (X2) (from (1.21)) in the above relation, we can see that

K =12f (X1 (X1f))− 21 (X1f)

2

16f2. (1.23)

Also, substituting (1.23) in (1.20), we get

f (X1 (X1f)) =7

4(X1f)

2 +4

3cf2 − f4.

From (1.22) it is easy to see that

∆f = −2∑i=1

(Xi (Xif)− (∇XiXi) f)

= −X1 (X1f) +3

4

(X1f)2

f.

Now, it is easy to see that

f∆f = −7

4(X1f)

2 − 4

3cf2 + f4 +

3

4(X1f)

2

= (X1f)2 − 4

3cf2 + f4.


Therefore, the mean curvature function of a non-CMC biconservative surface must

satisfy a second-order partial dierential equation

f∆f + | grad f |2 + 4

3cf2 − f4 = 0.

Remark 1.43. We note that in the above theorem obtained in [15], the authors did

not notice that grad f = 0 at any point of M implies f = 0 at any point.

Further, we can see that around any point of M there exist positively oriented local

coordinates (U ;u, v) such that f = f(u, v) = f(u) and (1.19) is equivalent to

ff ′′ − 7

4

(f ′)2 − 4

3cf2 + f4 = 0, (1.24)

i.e., f must satisfy a second-order ordinary dierential equation.

Indeed, let p0 ∈M be an arbitrary xed point of M and let γ = γ(u) be an integral

curve of X1 with γ(0) = p0. Let ϕ be the ow of X2 and (U ;u, v) positively oriented

local coordinates with p0 ∈ U such that

X(u, v) = ϕγ(u)(v) = ϕ(γ(u), v).

We have

Xu(u, 0) = γ′(u) = X1(γ(u)) = X1(u, 0)

and

Xv(u, v) = ϕ′γ(u)(v) = X2

(ϕγ(u)(v)

)= X2(u, v).

Of course, Xu, Xv is positively oriented. If we write the Riemannian metric g on M

in local coordinates as

g = g11du2 + 2g12dudv + g22dv

2,

we get g22 = |Xv|2 = |X2|2 = 1, and X1 can be expressed with respect to Xu and Xv as

X1 =1

σ(Xu − g12Xv) = σ gradu,

where σ =√g11 − g212 > 0, σ = σ(u, v).

Let f X = f(u, v). Since X2f = 0, we nd that

f(u, v) = f(u, 0) = f(u), (u, v) ∈ U.

It can be proved that

[X1, X2] =3 (X1f)

4fX2,

and therefore X2 (X1f) = X1 (X2f)− [X1, X2] f = 0.


On the other hand we have

X2 (X1f) = Xv

(1

σf ′)

= Xv

(1

σ

)f ′ (1.25)

= 0. (1.26)

We recall that

grad f = (X1f)X1 =

(1

σf ′)X1 = 0

at any point of U , and then f ′ = 0 at any point of U . Therefore, from equation (1.25),

Xv (1/σ) = 0, i.e., σ = σ(u). Since g11(u, 0) = 1, and g12(u, 0) = 0, we have σ = 1, i.e.,

on U

X1 = Xu − g12Xv = gradu. (1.27)

In [15] an equivalent expression for (1.19) it was found, i.e.,

f (X1X1f) =7

4(X1f)

2 +4c

3f2 − f4.

Therefore, using (1.27), relation (1.19) is equivalent to (1.24).

Remark 1.44. If φ : M2 → N3(c) is a non-CMC biharmonic surface, then, there

exists an open subset U such that grad f = 0 at any point of U , and f satises the

following system ∆f = f

(2c− |A|2

)A(grad f) = −f

2 grad f

,

on M . As we have seen, this system implies that, on U∆f = f

(2c− |A|2

)f∆f + | grad f |2 + 4

3cf2 − f4 = 0

,

which, in fact, is an ODE system. We getff ′′ − 3

4 (f′)2 + 2cf2 − 5

2f4 = 0

ff ′′ − 74 (f

′)2 − 43cf

2 + f4 = 0

. (1.28)

As an immediate consequence, we have(f ′)2

+10

3cf2 − 7

2f4 = 0,

and combining this with the rst integral(f ′)2

= 2f4 − 8cf2 + αf3/2


of the rst equation in (1.28), where α ∈ R is a constant, we obtain

3

2f5/2 +

14

3cf1/2 − α = 0.

If we denote f = f1/2, we get 3f5/2 + 14cf/3− α = 0. Thus, f satises a polynomial

equation with constant coecients, so f has to be a constant and then, f is also constant,

i.e., grad f = 0 on U (in fact, f has to be zero). Therefore, we come to a contradiction

and thus any biharmonic surface in N3(c) has to be CMC or minimal. From the rst

system we see that we can have proper-biharmonic surfaces in N3(c) only if c > 0 (see

[19,20] for c = 0, and [13,14] for c = ±1).

We can also note that relation (1.19) (which is extrinsic), together with (1.20), allows

us to nd an intrinsic relation that (M, g) must satisfy. More precisely, we have the

following result

Theorem 1.45 ([15]). Let φ :M2 → N3(c) be a biconservative surface with grad f = 0

at any point of M . Then the Gaussian curvature K satises

(i)

K = detA+ c = −3f2

4+ c; (1.29)

(ii) c−K > 0, gradK = 0 on M , and its level curves are circles in M with constant

curvature


;

(iii)

(c−K)∆K − | gradK|2 − 8

3K(c−K)2 = 0, (1.30)

where ∆ is the Laplace-Beltrami operator on M .

Proof. Formula (1.29) is just (1.20), which we have already proved.

In order to prove (ii), we rst note that from grad f = 0 on M one has f > 0 at

any point of M , as we have already seen, and from (1.29), we obtain c − K > 0 and

gradK = 0 at any point of M . We dene X1 = gradK/| gradK|. It is easy to see that

X1 = −X1, where X1 = grad f/| grad f |. Now, we dene the vector eld X2 = −X2

such thatX1, X2

is a positively oriented global frame eld.

We have already seen that X2f = 0. Thus, X2K = 0, i.e., the integral curves of X2

are the level curves of K. We note thatX2,−X1

is positively oriented.

From (1.22), it follows that

∇X2X2 = − 3X1K

8(c−K)X1 = −3| gradK|

8(c−K)X1


and[X1, X2

]K = X1

(X2K

)− X2

(X1K

)= −X2

(X1K

)=(∇X1

X2

)(K)−

(∇X2

X1

)(K) = (∇X1X2) (K)− (∇X2X1) (K).

Using (1.22), one obtains X2

(X1K

)= 0. Now, it is easy to see that we have

X2

(3(X1K

)/(8(c−K))

)= 0, i.e.,

κ =3X1K

8(c−K)

is constant along the level curves of K. If us consider Γ = Γ(v) an integral curve of X2,

i.e., X2(Γ(v)) = Γ′(v), for any v, since∣∣∣X2

∣∣∣ = 1, it follows that Γ is a curve parametrized

by arc-length. As Γ′,− X1

∣∣∣Γ is positively oriented, and

∇Γ′Γ′ = −κ|Γ X1

∣∣∣Γ

and ∇Γ′X1 = κ|ΓΓ′,

where κ|Γ is a constant, it follows that Γ is a circle with constant curvature κ.

The last item follows easily using (1.29) and (1.19).

Remark 1.46. Using the global orthonormal frame eld X1, X2 and the local co-

ordinates (u, v) from above, we can rewrite equation (1.30) as

24(c−K)X1 (X1K) + 33 (X1K)2 + 64K(c−K)2 = 0 (1.31)

or, equivalently,

24(c−K(u))K ′′(u) + 33(K ′(u)

)2+ 64K(u)(c−K(u))2 = 0. (1.32)

Remark 1.47. By standard transformations, equation (1.32) can be rewritten as a rst

order, nonhomogeneous, linear dierential equation and then, from the classical ODE

theory we can nd the rst integral(K ′)2 = 64

3K3 − 640

9cK2 + C(c−K)11/4 +

704

9c2K − 256

9c3,

where C ∈ R is a constant.

Corollary 1.48. Let φ :M2 → N3(c) be a biconservative surface. Assume that one of

the following holds

(i) c−K ≤ 0 on M ,

or


(ii) K is constant on M .

Then M is a CMC submanifold.

While the existence of biconservative immersions in 3-dimensional space forms with

the gradient of the mean curvature function dierent from zero at any point will be

proved in the next chapter, the uniqueness is treated here.

Theorem 1.49 ([27]). Let(M2, g

)be an abstract surface and c ∈ R a constant. If M

admits two biconservative immersions in N3(c) such that the gradients of their mean

curvature functions are dierent from zero at any point of M , then the two immersions

dier by an isometry of N3(c).

Proof. Let φ :(M2, g

)→ N3(c) be a biconservative immersion as in the statement of the

theorem. Then, we can assume that its mean curvature function is f = 2√(c−K)/3,

i.e., f depends only on the abstract surface(M2, g

). Dene X1 = grad f/| grad f |. We

have already seen that X1 = gradK/| gradK| satises X1 = −X1.

It is then easy to check that

A(X1

)= −

√c−K

3X1 and A

(X2

)=√3(c−K)X2,

whereX1, X2

is a positively oriented global orthonormal frame eld. It follows that

the shape operator of the immersion φ does not depend on φ, but only on the surface(M2, g

)and we can conclude.

Chapter 2Biconservative surfaces in

3-dimensional space forms

In this chapter, we focus on two issues: rst, we consider biconservative surfaces(M2, g

)in a 3-dimensional space formN3(c), with mean curvature function f satisfying grad f =0 at any point, and determine a certain Riemannian metric gr onM such that

(M2, gr

)is a Ricci surface in N3(c); second, we obtain the intrinsic necessary and sucient

conditions for an abstract surface to be locally embedded in N3(c) as a non-CMC

biconservative surface.

Most of the results in this chapter are original and they were presented in [27] and

[56]. There are also few results which are presented here for the rst time (see Theorem

2.10, Theorem 2.17, Theorem 2.21).

2.1 Biconservativity and minimality in N 3(c)

An abstract surface(M2, g

)with Gaussian curvature K is said to satisfy the Ricci

condition with respect to c (or simply the Ricci condition) if c−K > 0 and the metric√c−Kg is at, where c ∈ R is a constant. In this case,

(M2, g

)is called a Ricci

surface with respect to c (or simply a Ricci surface). As we will see further, in this

chapter, when c = 0, a surface satisfying the Ricci condition can be locally isometrically

embedded in R3 as a minimal surface. Actually, there exists a one-parameter family of

such embeddings. H. B. Lawson in [40] generalized this result by showing that the Ricci

condition is an intrinsic characterization of minimal surfaces in space forms N3(c), with

constant sectional curvature c (see also [65]).

In the following, we will see that the Ricci condition, as stated above, is equivalent

to an equation that looks very much like equation (1.30), satised by the Gaussian

curvature of a non-CMC biconservative surface in a space form N3(c). Then, a natural

27

28 Chapter 2. Biconservative surfaces in 3-dimensional space forms

question is whether there exists a simple way to transform surfaces satisfying (1.30) in

Ricci surfaces in N3(c). As it will turn out, the answer to this question is armative.

The following proposition points out some equivalent characterizations of Ricci sur-

faces.

Proposition 2.1. Let(M2, g

)be an abstract surface such that its Gaussian curvature

K satises c −K > 0, where c ∈ R is a constant. Then, the following conditions are

equivalent:

(i) K satises

(c−K)∆K − | gradK|2 − 4K(c−K)2 = 0; (2.1)

(ii) K satises

∆log(c−K) + 4K = 0; (2.2)

(iii) the metric√c−Kg is at.

Moreover, when c = 0, we also have a fourth equivalent condition:

(iv) the metric (−K)g has constant Gaussian curvature equal to 1.

Proof. First, we prove that

∆log(c−K) =(K − c)∆K + | gradK|2

(c−K)2. (2.3)

Let us consider p ∈M an arbitrary point and X1, X2 a local orthonormal frame eld

that is geodesic around p. Then, at p we have

∆log(c−K) =−2∑i=1

Xi (Xi log(c−K))

=− ∆K

c−K+

2∑i=1

(Xi(K))2

(c−K)2

=− ∆K

c−K+

| gradK|2

(c−K)2

=(K − c)∆K + | gradK|2

(c−K)2.

Obviously, since p was arbitrary xed in M , relation (2.3) holds globally.

Now it is easy to see that (i) and (ii) are equivalent. If we assume that (i) holds and

we substitute (c−K)∆K from (2.3) in (2.1), we nd (2.2). Conversely, if (ii) holds, we

substitute ∆log(c−K) from (2.2) in (2.3) and obtain (2.1).

2.1. Biconservativity and minimality in N3(c) 29

Next, in the same way as in [53], we consider a family of Riemannian metrics on M

given by gr = (c −K)rg, where r ∈ R is a constant. From equation (1.6), one obtains

that the Gaussian curvature curvature Kr of gr is given by

Kr = (c−K)−r(K +

1

2∆ log(c−K)r

). (2.4)

If (ii) holds, then substituting (2.2) in (2.4), we get Kr = (1− 2r)(c−K)−rK. Now, if

we consider the particular cases, r = 1/2 or r = 1, it follows that (ii) implies (iii) and

(iv). Conversely, it is easy to see, from (2.4), that (iii) implies (ii) and also, if c = 0,

(iv) implies (ii).

Remark 2.2. Proposition 2.1 was rst proved in the case when c = 0 in [53].

Working exactly as in the proof of Proposition 2.1 we get the following result.


)be an abstract surface such that its Gaussian curvature

K satises c −K > 0, where c ∈ R is a constant. Then, the following conditions are

equivalent:

(i) K satises equation (1.30);

(ii) ∆log(c−K) + 83K = 0;

(iii) the metric (c−K)3/4g is at.

Moreover, when c = 0, we also have a fourth equivalent condition:

(iv) the metric (−K)g has constant Gaussian curvature equal to 1/3.

Now, we can state our rst main result.

Theorem 2.4. Let(M2, g

)be an abstract surface with negative Gaussian curvature K

that satises

K∆K + | gradK|2 + 8

3K3 = 0. (2.5)

Then(M2,

√−Kg

)is a Ricci surface in R3.

Proof. From Proposition 2.1, one can see that suces to show that there exists a

Riemannian metric on M , conformally equivalent to g, that satises (2.2).

In order to nd such a metric, let us consider again the metrics gr = (−K)rg, with

r ∈ R. Since K satises (2.5), and therefore equation (1.30) for c = 0, from Proposition

2.3, it follows that

∆log(−K) = −8

3K. (2.6)


Now, substituting (2.6) in equation (2.4) corresponding to c = 0, one obtains that the

Gaussian curvature curvature Kr of gr is given by

Kr = (−K)−r(K +

1

2∆ log(−K)r

)= −3− 4r

3(−K)1−r.

Assume that 3− 4r > 0, i.e., Kr < 0, and then, using equations (1.7) and (2.6), we can

compute

∆r log(−Kr) = ∆r log

(3− 4r

3(−K)1−r

)= (1− r)∆r log(−K)

= (1− r)(−K)−r∆log(−K)

=8(1− r)

3(−K)1−r,

where ∆r is the Laplacian of gr. Now, equation (2.2) becomes

0 = ∆r log(−Kr) + 4Kr

=8(1− r)

3(−K)1−r − 4(3− 4r)

3(−K)1−r =

4(2r − 1)

3(−K)1−r

and we get that r = 1/2.

We just have proved that(M2, g1/2 =

√−Kg

)is a Ricci surface with Gaussian

curvature K1/2 = −(1/3)√−K < 0.

From Theorem 1.45 and Theorem 2.4, one obtains the following corollary.

Corollary 2.5. Let(M2, g

)be a biconservative surface in R3, where g is the induced

metric on M . If (grad f)(p) = 0 at any point p ∈ M , then(M2,

√−Kg

)is a Ricci

surface.

Remark 2.6. In the same way as in Theorem 2.4, one can show that if(M2, g

)is a

Ricci surface in R3 with negative Gaussian curvature K, then the Gaussian curvature

of(M2, (−K)−1g

)is negative and satises equation (2.5).

Although the method used to prove Theorem 2.4 does not work in the case of non-

at space forms, it is still possible to extend this result to the case of space forms, as

shown by the following theorem.


)be a biconservative surface in a space form N3(c) with

induced metric g and Gaussian curvature K. If (grad f)(p) = 0 at any point p ∈ M ,

then, on an open dense subset,(M2, (c−K)rg

)is a Ricci surface in N3(c), where r is

a locally dened function that satises

K +∆

(1

4log (c−Kr) +

r

2log(c−K)

)= 0, (2.7)


with the Gaussian curvature Kr of (c−K)rg given by

Kr = (c−K)−r(3− 4r

3K +

1

2log(c−K)∆r + (c−K)−1g(grad r, gradK)

).

Proof. Let us consider a family of Riemannian metrics gr = (c−K)rg on M , this time

r being a function on M . From (1.6), we have that the Gaussian curvature Kr of gr is

given by

Kr = (c−K)−r(K +

1

2∆(r log(c−K))

), (2.8)

where K is the Gaussian curvature of g. Since M is biconservative, then relation (1.30)

holds, and using Proposition 2.3, and the well-known property (see, for example [17])

∆(αβ) = (∆α)β + α(∆β)− 2g(gradα, gradβ),

where α, β are smooth functions on M , we obtain that

∆(r log(c−K)) = r∆log(c−K) + log(c−K)∆r − 2g(grad r, grad log(c−K))

= −83rK + log(c−K)∆r + 2(c−K)−1g(grad r, gradK),

(2.9)

If we substitute the above expression in (2.8) it follows that

Kr = (c−K)−r(3− 4r

3K +

1

2log(c−K)∆r + (c−K)−1g(grad r, gradK)

). (2.10)

Next, assume that (c−Kr) (p) > 0 at any point p ∈M and consider a new Riemannian

metric g on M given by

g =√c−Krgr =

√c−Kr(c−K)rg

= e2ρg.

From the denition of g, one obtains

ρ =1

4log (c−Kr) +

r

2log(c−K). (2.11)

We denote by K the Gaussian curvature corresponding to g and ask that it vanishes.

From equation (1.6) we have

K =√c−Kr(c−K)−r(K +∆ρ).

Therefore, K = 0 is equivalent to

K +∆ρ = 0. (2.12)


Now, if we consider a (arbitrary) local orthonormal frame eld X1, X2 on M , we can

see that

∆log (c−Kr) =−2∑i=1

Xi (Xi (log (c−Kr)))− (∇XiXi) (log (c−Kr))

=−2∑i=1

(c−Kr)

−1 (−Xi (Xi (Kr)) + (∇XiXi) (Kr))

− (c−Kr)−2 (Xi (Kr))

2

=(c−Kr)

−2((Kr − c)∆Kr + |gradKr|2

). (2.13)

Using (2.11), (2.13), (2.9), and then (2.10) we get on M

∆ρ =1

4∆ log (c−Kr) +

1

2∆(r log(c−K))

=1

4(c−Kr)

−2 ((Kr − c)∆Kr + | gradKr|2)−4

3rK +

1

2log(c−K)∆r

+ (c−K)−1g(grad r, gradK)

=1

4(c−Kr)

−2

(Kr − c)∆

((c−K)−r

(3− 4r

3K +

1

2log(c−K)∆r


))+

∣∣∣∣ grad((c−K)−r(3− 4r

3K +

1

2log(c−K)∆r


))∣∣∣∣2− 4

3rK +

1

2log(c−K)∆r + (c−K)−1g(grad r, gradK).

Equation (2.12) is a fourth order PDE in r. The leading term is

−1

8(c−Kr)

−1 log(c−K)∆2r

and all the other derivatives of r are of order less or equal to three.

Now, let us consider X1 = (grad f)/| grad f |. Since grad f = 0 at any point of M ,

then X1, X2 is a global orthonormal frame eld on M , as in Theorem 1.42. We recall

that, in the same theorem, it was proved that X2f = 0, which implies, using (1.20),

that also X2K = 0.

Assuming that r is a function on M such that X2r = 0, from formulas in (1.22),

it easily follows that [X1, X2] (K) = 0 and [X1, X2] (r) = 0. Therefore, we also have

X2 (X1K) = 0 and X2 (X1r) = 0.

Obviously, from (1.29), we have gradK = 0 at any point of M . We note that the

function log(c−K) cannot vanish on an open subset of M . Indeed, if we assume that


log(c−K) = 0 on an open subset, it follows that grad(log(c−K)) = 0 on that subset,

where also gradK = 0, which is a contradiction.

Now, away from the points where log(c −K) = 0, using the above equation, then

(1.22) and (1.29), equation (2.12) can be written as

∆2r = F (r,X1r,X1(X1r), X1(X1(X1r))), (2.14)

where the coecients in the expression F in the right hand side are smooth functions

depending on K, X1(K), X1(X1K), and X1(X1(X1K)). In fact, these coecients

depend only on K and X1K, as (1.31) holds.

Let us consider a point p0 ∈ M , where log(c − K) = 0, and γ = γ(u) an integral

curve of X1 with γ(0) = p0. Let ϕ be the ow of X2 and, on a neighborhood U ⊂M of

p0, dene a local parametrization of M ,

X(u, v) = ϕγ(u)(v) = ϕ(γ(u), v).

We have X(u, 0) = ϕγ(u)(0) = γ(u),

Xu(u, 0) = ∂u(u, 0) = γ′(u) = X1(γ(u)) = X1(u, 0),

and

Xv(u, v) = ∂v(u, v) = ϕ′γ(u)(v) = X2(ϕγ(u)(v)) = X2(u, v),

for any u and v. Assume now that u ∈ I, where I is an open interval containing 0 such

that log(c−K(u)) = 0, for any u ∈ I.

By hypothesis, we have X2r = 0, which means that r = r(u) on U . Moreover, from

(1.27), we have X1 = Xu − g12Xv. Thus, on U , X1(X1r) = r′′(u), X1 (X1 (X1r)) =

r′′′(u), and X1 (X1 (X1 (X1r))) = r(iv)(u), respectively. Moreover, the same formulas

hold if we take K instead of r. Therefore, on U , equation (2.14) becomes

r(iv)(u) = F (u, r(u), r′(u), r′′(u), r′′′(u)). (2.15)

We note that the coecients in the expression of F in the right hand side are smooth

functions depending only on K and K ′(u), as shown by (1.32).

The initial conditions follow from (c−Kr) (p0) > 0, i.e., from

(c−K(0))−r(0)(3− 4r(0)

3K(0) +

1

2log(c−K(0))

(− r′′(0) +

3f ′(0)

4f(0)r′(0)

)+ (c−K(0))−1r′(0)K ′(0)

)< c.

We can choose r(0), r′(0), and r′′(0) such that the above inequality is satised. Indeed,

consider a smooth function H : I × R3 → R dened by

H(u, ξ1, ξ2, ξ3

)=(c−K(u))−ξ

1

(3− 4ξ1

3K(u)

+1

2log(c−K(u))

(− ξ3 +

3f ′(u)

4f(u)ξ2)+ (c−K(u))−1ξ2K ′(u)

).


Then, as the limit of H, when(u, ξ1, ξ2, ξ3

)goes to 0, is K(0) which is less than c, then

there exists an open subset D ⊂ R4 containing 0,

D = (−ε, ε)× (−ε, ε)× (−ε, ε)× (−ε, ε) = (−ε, ε)4 ,

with (−ε, ε) ⊂ I, such that H(u, ξ1, ξ2, ξ3

)< c, for any

(u, ξ1, ξ2, ξ3

)∈ D.

Now, from the ODE's theory, we know that equation (2.15), with the given initial

conditions of type (0, r0, r

′0, r

′′0 , r

′′′0

)∈ D × R,

has a unique solution. More precisely, we denote

ξ1 = r(u), ξ2 = r′(u), ξ3 = r′′(u), ξ4 = r′′′(u)

and consider

G : D × R = (−ε, ε)× (−ε, ε)3 × R → R4,

dened by

G(u, ξ1, ξ2, ξ3, ξ4

)=(ξ2, ξ3, ξ4, F

(u, ξ1, ξ2, ξ3, ξ4

)).

Equation (2.15) is equivalent to

ξ′(u) = G

(u, ξ(u)

). (2.16)

The function G is smooth and therefore, for the initial condition(0, r0, r

′0, r

′′0 , r

′′′0

)∈ D × R,

there exists a unique local solution ξ = ξ(u) dened around 0. Thus, equation (2.15)

has a unique local solution r = r(u).

This means that there exists a at Riemannian metric g =√c−Krgr on (a smaller)

U , and then we use [40, Theorem 8] to conclude that, on the open dense subset, our

surface (M2, g) can be locally conformally embedded in N3(c) as a minimal surface.

Remark 2.8. It is straightforward to verify that, when c = 0, the only constant solution

of equation (2.7) is r = 1/2. When c > 0, we note that r = 3/4 is a solution of (2.7).

Therefore,(M2, (c−K)3/4g

)is a at surface and then, trivially, a Ricci surface with

respect to c > 0; it can be immersed in the Euclidean 3-dimensional sphere of radius

1/√c as the minimal Cliord torus.

Remark 2.9. As we will see in the next section (see Theorem 2.18), the hypotheses of

Theorem 2.7 can be replaced by:

Let(M2, g

)be an abstract surface and c ∈ R a constant. Assume that c−K > 0,

gradK = 0 at any point of M , and the level curves of K are circles in M with constant

curvature


.

2.2. An intrinsic characterization of biconservative surfaces in N3(c) 35

2.2 An intrinsic characterization of biconservative

surfaces in N 3(c)

While any of the equivalent conditions in Proposition 2.1 characterizes intrinsically min-

imal surfaces in 3-dimensional space forms N3(c), the similar conditions in Proposition

2.3 alone fail to do the same in the case of biconservative surfaces. In this section, we

will nd the intrinsic necessary and sucient conditions for an abstract surface to be

locally embedded in N3(c) as a non-CMC biconservative surface.

According to Theorem 1.45, (ii), we present some conditions equivalent to the fact

that the level curves of the Gaussian curvature are circles.


)be an abstract surface with Gaussian curvature K satis-

fying c−K(p) > 0 and (gradK)(p) = 0 at any point p ∈M , where c ∈ R is a constant.

Let X1 = (gradK)/| gradK| and X2 ∈ C(TM) be two vector elds on M such that

X1(p), X2(p) is a positively oriented basis at any point p ∈ M . Then, the following

conditions are equivalent:

(i) the level curves of K are circles in M with constant curvature


=3X1K

8(c−K);

(ii)

X2 (X1K) = 0 and ∇X2X2 =−3X1K

8(c−K)X1;

(iii)

∇X1X1 = ∇X1X2 = 0, ∇X2X2 = − 3X1K

8(c−K)X1, ∇X2X1 =

3X1K

8(c−K)X2.

Proof. In order to prove (i) ⇒ (ii), we note that from the denition of X1, X2 we

have X2K = 0, i.e., the integral curves of X2 are the level curves of K. Since κ is

constant along the integral curves of X2, one obtains X2 (X1K) = 0. Let us now

consider Γ = Γ(v) an integral curve of X2, i.e., X2(Γ(v)) = Γ′(v), for any v. It follows

that Γ is a circle in M with curvature κ. Since |X2| = 1, we also get that Γ is a curve

parametrized by arc-length. As Γ′,− X1|Γ is positively oriented, one obtains

∇Γ′Γ′ = ∇X2X2 = −κ|Γ X1|Γ .

The proof of (ii) ⇒ (i) can be found in Theorem 1.45, (ii).

To prove that (iii) implies (ii) it is enough to note that [X1, X2]K = −X2 (X1K)

and also [X1, X2]K = − (∇X2X1) (K) = 0. Therefore, we have X2 (X1K) = 0.


In order to show the converse implication, i.e., (ii) implies (iii), we note that from the

expression of∇X2X2, one obtains, as we have already seen,∇X2X1 = 3 (X1K)X2/(8(c−K)). Thus, [X1, X2]K = (∇X1X2) (K) = 0, which is equivalent to ⟨∇X1X2, gradK⟩ =0, i.e., ⟨∇X1X2, X1⟩ = 0. Then ∇X1X2 = 0. Now, it is easy to see that ∇X1X1 =

−⟨X1,∇X1X2⟩ = 0.

Remark 2.11. The integral curves of X2 are circles in M with constant curvature

κ =3X1K

8(c−K)=

3| gradK|8(c−K)

and the integral curves of X1 are geodesics of M .

The next result gives a description of the metrics for which the level curves of K are

circles.



fying (gradK)(p) = 0 and c−K(p) > 0 at any point p ∈M , where c ∈ R is a constant.


X1(p), X2(p) is a positively oriented orthonormal basis at any point p ∈ M . If the

level curves of K are circles in M with constant curvature

κ =3X1K

8(c−K)=

3| gradK|8(c−K)

,

then, for any point p0 ∈ M , there exists a positively oriented parametrization X =

X(u, v) of M in a neighborhood U ⊂M of p0 such that

(i) the curve u → X(u, 0) is an integral curve of X1 with X(0, 0) = p0 and v →X(u, v) is an integral curve of X2, for any u and v;

(ii) K(u, v) = (K X)(u, v) = (K X)(u, 0) = K(u), for any (u, v);

(iii) for any pair (u, v), we have

g11(u, v) =9

64

(K ′(u)

c−K(u)

)2

v2 + 1,

g12(u, v) = − 3K ′(u)

8(c−K(u))v, g22(u, v) = 1;

(iv) the Gaussian curvature K = K(u) satises

24(c−K)K ′′ + 33(K ′)2 + 64K(c−K)2 = 0.


Proof. Let p0 be a xed point in M , γ = γ(u) an integral curve of X1, i.e., X1(γ(u)) =

γ′(u), with γ(0) = p0, and ϕ the ow of X2. Consider again

X(u, v) = ϕγ(u)(v) = ϕ(γ(u), v).

As we have already seen, X(u, 0) = ϕγ(u)(0) = γ(u),

Xu(u, 0) = ∂u(u, 0) = γ′(u) = X1(γ(u)) = X1(u, 0),

and

Xv(u, v) = ∂v(u, v) = ϕ′γ(u)(v) = X2(ϕγ(u)(v)) = X2(u, v),

for any u and v.

Since Xv(u, v) = X2(u, v), it follows that |Xv(u, v)|2 = 1, which means that

g22(u, v) = 1. (2.17)

We also have, for any u,

g11(u, 0) = |Xu(u, 0)|2 = 1 and g12(u, 0) = g (Xu(u, 0), Xv(u, 0)) = 0. (2.18)

Next, we nd the expression of X1 with respect to Xu = ∂u and Xv = ∂v. We

write X1 = α∂u + β∂v, where α and β are smooth functions such that α(u, 0) = 1 and

β(u, 0) = 0. Using (2.17), it follows that

1 = g (X1, X1) = α2g11 + 2αβg12 + β2g22 = α2g11 + 2αβg12 + β2,

and

0 = g (X1, X2) = αg12 + βg22 = αg12 + β.

From the second equation, one obtains β = −αg12 and, replacing in the rst one, we

get 1 = α2(g11 − g212

). Let us denote σ(u, v) =

√g11 − g212 > 0 and then we have

X1 =1

σ(∂u − g12∂v) . (2.19)

It is easy to see that α = 1/σ and β = −g12/σ. Next, we note that, from the denition

of X1 and X2, since X1(K) = | grad(K)|, one obtains X2K = 0, i.e., the integral curves

v → ϕγ(u)(v) of X2 are the level curves of K, which means that v → K(ϕγ(u)(v)) is a

constant function. Also, identifying K with K X, we can write K = K(u, v). Since

X2K = 0, it actually follows that K(u, v) = K(u, 0) = K(u), for any pair (u, v). The

level curves v → ϕγ(u)(v) of K are parametrized by arc-length and, by hypothesis, are

circles with constant curvature κ = 3X1K/(8(c −K)), which means, also using (2.19)

and the fact that κ(u, v) = κ(u), that

X1K =8

3κ(c−K) =

8

3κ(u)(c−K(u)) (2.20)

=1

σK ′ =

1

σ(u, v)K ′(u),


which implies that X2 (X1K) = 0 and

σ(u, v) =K ′(u)

8κ(u)(c−K(u))= σ(u) = 1,

for any u and v. Hence X1 = ∂u − g12∂v = gradu.

Now, let us x the parameter u. As X2,−X1 is positively oriented, we have

∇ϕ′γ(u)

(v)ϕ′γ(u)(v) = ∇X2X2 = κ (−X1)

= Γ122∂u + Γ2

22∂v

and then

κ = g(∇ϕ′

γ(u)(v)ϕ

′γ(u)(v),−X1

)= g

(Γ122∂u + Γ2

22∂v,−∂u + g12∂v)

= −Γ122

(g11 − g212

)= −Γ1

22,

where Γkij are the Christoel symbols. Thus, κ(u) = −Γ122(u, v), for any v, and for any

u.

Using the denition of Γ122 and 1 = σ2 = g11 − g212, we have, for any u and v

Γ122 =

1

2g11(∂g21∂v

+∂g21∂v

− ∂g22∂u

)+

1

2g12(∂g22∂v

+∂g22∂v

− ∂g22∂v

)= g11

∂g12∂v

=∂g12∂v

.

So, κ = −∂g12∂v . From equation (2.20), it follows that

K ′(u) = −8

3

∂g12∂v

(c−K(u)),

for any u and v, which leads to

∂g12∂v

= − 3K ′(u)

8(c−K(u))=

3

8(log(c−K(u)))′

and, therefore,

g12(u, v) = − 3K ′(u)

8(c−K(u))v + θ(u).

But, from (2.18), we know that g12(u, 0) = 0, which implies θ(u) = 0, and we conclude

with

g12(u, v) = − 3K ′(u)

8(c−K(u))v, (2.21)

for any u and v.

Finally, since 1 = σ2 = g11 − g212, we nd

g11(u, v) =9

64

(K ′(u)

c−K(u)

)2

v2 + 1. (2.22)


By a straightforward computation one gets the expressions of the Christoel symbols

Γ111 = −33

29

(K′(u)c−K(u)

)3v2, Γ1

12 = Γ121 = −Γ2

22 =32

26

(K′(u)c−K(u)

)2v,

Γ212 = Γ2

21 =33

29

(K′(u)c−K(u)

)3v2, Γ1

22 = − 323

K′(u)c−K(u)

(2.23)

and

Γ211 = − 3

23

(33

29

(K ′(u)

c−K(u)

)4

v3 +K ′′(u)

c−K(u)v +

11

23

(K ′(u)

c−K(u)

)2

v

). (2.24)

Using these expressions, we reobtain the formulas for the Levi-Civita connection

∇X1X1 = ∇X1X2 = 0, ∇X2X2 = − 3X1K

8(c−K)X1, ∇X2X1 =

3X1K

8(c−K)X2.

Now, from (2.23) and (2.24), one sees, after a straightforward computation, that the

Gauss equation of(M2, g

)is

K = − 1

g11

(Γ212

)u−(Γ211

)v+ Γ1

12Γ211 + Γ2

12Γ212 − Γ2

11Γ222 − Γ1

11Γ212

,

where(Γkij

)u=

∂Γkij

∂u , is equivalent to

24(c−K)K ′′ + 33(K ′)2 + 64K(c−K)2 = 0. (2.25)

Remark 2.13. Using equation (2.25), we can rewrite (2.24) in a simpler way as

Γ211 =

34

212

(K ′(u)

c−K(u)

)4

v3 +Kv.

Remark 2.14. It is easy to verify that, in the hypotheses of Theorem 2.12, equation

24(c−K)K ′′ + 33(K ′)2 + 64K(c−K)2 = 0

can be written as

(c−K)∆K − | gradK|2 − 8

3K(c−K)2 = 0.

Remark 2.15. Considering the change of coordinates (u, v) →(u, (c−K)3/8v

)= (u, s) in Theorem 2.12, we obtain, after a straightforward computation, a simpler

expression

g = du2 + (c−K)−3/4ds2


for the Riemannian metric on the surface. Moreover, if we consider a second change of

coordinates (u, s) →(∫ u

u0(c−K(τ))3/8 dτ, s

)= (u, s), then the metric g can be written

as

g = (c−K(u))−3/4(du2 + ds2

),

where K(u) = K(u(u)), which means that (u, s) are isothermal coordinates on the

surface.

The converse of Theorem 2.12 is the following result that ensures the existence of

surfaces(M2, g

)such that the level curves of the Gaussian curvature are circles. It can

be proved by a straightforward computation.

Theorem 2.16. Let D be an open subset of R2 = Ouv and c ∈ R a constant. Consider

K = K(u) a function on D such that K ′(u) > 0 and c−K(u) > 0, for any u, and

24(c−K)K ′′ + 33(K ′)2 + 64K(c−K)2 = 0.

Dene a Riemannian metric g = g11du2 + 2g12dudv + g22dv

2 on D by

g11(u, v) =9

64

(K ′(u)

c−K(u)

)2

v2 + 1,

g12(u, v) = − 3K ′(u)

8(c−K(u))v, g22(u, v) = 1.

Then, K is the Gaussian curvature of g and its level curves v → (u0, v) are circles in

(D, g) with curvature κ = 3K ′(u)/(8(c−K(u))).


)be an abstract surface and c ∈ R a constant. Assume that

c−K > 0 and gradK = 0 at any point of M , and the level curves of K are circles in

M with constant curvature


.

If there exists a biconservative immersion φ :(M2, g

)→ N3(c), then grad f = 0, f > 0

at any point of M , and φ is unique.

Proof. Assume that there exists a biconservative immersion φ :(M2, g

)→ N3(c). First,

we prove that grad f = 0 on an open dense subset of M . Indeed, if we assume that

W = p ∈M | (grad f)(p) = 0

is not dense, then we have that grad f = 0 on M \W , which is an open, non-empty set.

Let us denote by V a connected component of M \W . We note that V is also open in

M . In Remark 2.14, we have seen that K satisfy

24(c−K)K ′′ + 33(K ′)2 + 64K(c−K)2 = 0. (2.26)


On the other hand, as φ is CMC on V , using the local coordinates (u, v) as above,

equation (1.12) can be rewritten as

8(c−K)K ′′ +

(8(c−K)

|H|2 + c−K+ 3

)(K ′)2 + 32K(c−K) = 0. (2.27)

Combining equations (2.26) and (2.27), one obtains

3|H|2(K ′)2 − 4K(c−K)

((|H|2 + c−K

)2+ 2|H|2

(|H|2 + c−K

))= 0. (2.28)

Now, we recall that in Remark 1.47 is given a rst integral of

24(c−K)K ′′ + 33(K ′)2 + 64K(c−K)2 = 0.

The rst integral is(K ′)2 = 64

3K3 − 640

9cK2 + C(c−K)11/4 +

704

9c2K − 256

9c3,

where C ∈ R is a constant.

Substituting (K ′)2 in (2.28), we obtain that K has to satisfy a fourth order polyno-

mial equation with constant coecients, with the leading term 4K4, so K has to be a

constant and this is a contradiction.

Thus, grad f = 0 on W , which is an open dense subset of M . From the Gauss

equation, K = c+ detA, we obtain on W

f2 =4

3(c−K).

As W is dense in M , it follows that, in fact, the above relation holds on whole M .

Therefore, since c−K > 0 and gradK = 0 on M , one obtains f > 0 and grad f = 0 at

any point of M .

Finally, the uniqueness of φ follows from Theorem 1.49.

We are now ready to prove the main result of this section, which provides an in-

trinsic characterization of non-CMC biconservative surfaces in a 3-dimensional space

form N3(c). Basically, the intrinsic conditions given in Theorem 1.45, (ii), ensure the

existence of a non-CMC biconservative immersion in N3(c).

Theorem 2.18. Let (M2, g) be an abstract surface and c ∈ R a constant. Then M can

be locally isometrically embedded in a space form N3(c) as a biconservative surface with

the gradient of the mean curvature dierent from zero at any point p ∈M if and only if

the Gaussian curvature K satises c−K(p) > 0, (gradK)(p) = 0, and its level curves

are circles in M with constant curvature


.


Proof. The direct implication was proved in Theorem 1.45, (ii).

To prove the converse, let us consider X1 = (gradK)/| gradK| and X2 ∈C(TM) two vector elds such that X1(p), X2(p) is a positively oriented orthonor-

mal basis at any point p ∈ M . From Theorem 2.12 we have seen that the Levi-Civita

connection on (M2, g) is given by

∇X1X1 = ∇X1X2 = 0, ∇X2X2 = − 3X1K

8(c−K)X1, ∇X2X1 =

3X1K

8(c−K)X2.

Now, consider f = (2/√3)√c−K > 0 and, since X2K = 0, we easily get

grad f = − X1K√3(c−K)

X1 = − gradK√3(c−K)

.

Dene X1 = (grad f)/| grad f | = −X1 and X2 = −X2 and then

∇X1X1 = ∇X1

X2 = 0,

and

∇X2X2 = ∇X2X2 = − 3X1K

8(c−K)X1, ∇X2

X1 = ∇X2X1 =3X1K

8(c−K)X2.

Since (X1f)/f = −(X1K)/(2(c−K)), one obtains

∇X2X2 =

3X1f

4fX1 and ∇X2

X1 = −3X1f

4fX2.

Let us now consider a tensor eld A of type (1, 1) on M dened by

AX1 = −f2X1 and AX2 =

3f

2X2.

It is straightforward to verify that A satises the Gauss equation

K = c+ detA

and the Codazzi equation

(∇X1A)X2 = (∇X2

A)X1,

which means that the surface M can be locally isometrically embedded in N3(c) with

A as its shape operator. Moreover, from the denition of A, it is easy to see that

A(grad f) = −f2grad f,

which shows, using (1.16), that M is a biconservative surface in N3(c).


Remark 2.19. If we consider the local coordinates (u, v) corresponding to the frame

eld X1, X2, as in the proof of Theorem 2.12, then (−u,−v) represent the local

coordinates corresponding to the frame eldX1, X2

, as we have seen in the proof of

(1.24).

Remark 2.20. If the surfaceM in Theorem 2.18 is simply connected, then the theorem

holds globally, but, in this case, instead of a local isometric embedding we have a global

isometric immersion.

We note that, unlike in the minimal immersions case, if M satises the hypotheses

in Theorem 2.18, then there exists a unique biconservative immersion in N3(c) (up to

an isometry of N3(c)), and not a one-parameter family.

Using Theorem 2.17 and Theorem 2.18 we can state the following result.


)be an abstract surface and c ∈ R a constant. Assume that

c−K > 0 and gradK = 0 at any point of M , and the level curves of K are circles in

M with constant curvature


.

Then, there exists a unique biconservative immersion φ :(M2, g

)→ N3(c). Moreover,

the gradient of its mean curvature function is dierent from zero at any point of M .

Using local isothermal coordinates, we can nd some more intrinsic characteriza-

tions of biconservative surfaces in N3(c). These characterizations give some explicit

expressions for the metric g.



fying c−K(p) > 0 and (gradK)(p) = 0 at any point p ∈M , where c ∈ R is a constant.


X1(p), X2(p) is a positively oriented basis at any point p ∈ M . Then, the following

conditions are equivalent:

(i) the level curves of K are circles in M with constant curvature


=3X1K

8(c−K);

(ii) the metric g can be written locally, as g = (c −K)−3/4(du2 + dv2

), where (u, v)

are local coordinates positively oriented, K = K(u), and K ′ > 0;

(iii) the metric g can be written locally, as g = e2ρ(du2 + dv2

), where (u, v) are local

coordinates positively oriented, and ρ = ρ(u) satises the equation

ρ′′ = e−2ρ/3 − ce2ρ (2.29)


and the condition ρ′ > 0; moreover, the solutions of the above equation, u = u(ρ),

are

u =

∫ ρ

ρ0

dτ√−3e−2τ/3 − ce2τ + a

+ u0,

where ρ is in some open interval I, ρ0 ∈ I and a, u0 ∈ R are constants;

(iv) the metric g can be written locally, as g = e2ρ(du2 + dv2

), where (u, v) are local

coordinates positively oriented, and ρ = ρ(u) satises the equation

3ρ′′′ + 2ρ′ρ′′ + 8ce2ρρ′ = 0 (2.30)

and the conditions ρ′ > 0 and c+e−2ρρ′′ > 0; moreover, the solutions of the above

equation, u = u(ρ), are

u =

∫ ρ

ρ0

dτ√−3be−2τ/3 − ce2τ + a

+ u0,

where ρ is in some open interval I, ρ0 ∈ I and a, b, u0 ∈ R are constants, b > 0.

Proof. The implication (i) ⇒ (ii) was already proved in Remark 2.15.

To prove (ii) ⇒ (i), consider a local orthonormal frame eld positively oriented

Y1, Y2 by Y1 = (c−K)3/8∂u and Y2 = (c−K)3/8∂v. It is easy to see that gradK = (c−K)3/8K ′(u)Y1 and, then | gradK| = (c −K)3/8K ′(u). Since X1 = (gradK)/| gradK|,we obtain X1 = Y1 and, therefore X2 = Y2. By direct computation, from the denitions

of Y1 and Y2 one gets

X2 (X1K) = Y2 (Y1K) = (c−K)3/8∂v

((c−K)3/8K ′

)= 0

and

∇X2X2 = ∇Y2Y2 = (c−K)3/4∇∂v∂v.

Further, to write ∇∂v∂v with respect to ∂u and ∂v, we compute the Christoel symbols

Γ122 = − 3

8(c−K)K ′ and Γ2

22 = 0

and obtain

∇∂v∂v = Γ122∂u + Γ2

22∂v = − 3K ′

8(c−K)∂u.

Therefore

∇X2X2 = − 3K ′

8(c−K)1/4∂u = −3(c−K)3/8K ′

8(c−K)(c−K)3/8∂u

= − 3Y1K

8(c−K)Y1 = − 3X1K

8(c−K)X1.

Now we use Theorem 2.10 to conclude.


To prove that (ii) implies (iii), consider a smooth function ρ = ρ(u) such that

(c −K)−3/4 = e2ρ. It follows that K = c − e−8ρ/3. On the other hand, from (1.6) we

have K = −e−2ρρ′′. Therefore,

ρ′′ = e−2ρ/3 − ce2ρ (2.31)

and K = −e−8ρ/3 + c. Since K ′ = 8ρ′e−8ρ/3/3 > 0, one obtains ρ′ > 0.

In order to solve equation (2.31), rst we multiply the equation by 2ρ′ and then,

integrating, one obtains (ρ′)2

= −3e−2ρ/3 − ce2ρ + a,

where a ∈ R is a constant. Since ρ′ > 0, we get

dρ

du=√−3e−2ρ/3 − ce2ρ + a,

which leads to

u =

∫ ρ

ρ0

dτ√−3e−2τ/3 − ce2τ + a

+ u0,

where ρ is in some open interval I, ρ0 ∈ I and u0 ∈ R is a constant.

To prove (iii) ⇒ (ii), dene ρ = −(3 log(c−K))/8 and, we only have to show that

this function satises equation (2.29) and ρ′ > 0 if and only if K ′ > 0. As we have

seen the metric g can be written as g = (c−K)−3/4(du2 + dv2

). By a straightforward

computation, from (1.6), one obtains

K = −e−2ρρ′′

= −e−2ρ3(K ′′(c−K) + (K ′)2

)8(c−K)2

= −3(K ′′(c−K) + (K ′)2

)8(c−K)5/4

.

It follows that K has to satisfy

3K ′′(c−K) + 3(K ′)2 + 8K(c−K)5/4 = 0.

Using the above relation it is easy to see that ρ = −(3 log(c − K))/8 is a solution of

(2.29).

From the denition of ρ, one gets that ρ′ > 0 if and only if K ′ > 0.

In order to show (iii) ⇒ (iv), consider the change of coordinates (u, v) = (αu, αv),

where α > 0 is a real constant. Then, the metric g can be written as

g = e2(ρ(u)+logα)(du2 + dv2

),


where ρ(u) = ρ (u (u)). We denote ϕ (u) = ρ(u) + logα and by a direct computation

obtain that (2.29) is equivalent to

ϕ′′ (u) = α8/3e−2ϕ(u)/3 − ce2ϕ(u). (2.32)

Multiplying the above relation by e2ϕ(u)/3 and then dierentiating, we get that ϕ satises

3ϕ′′′ (u) + 2ϕ′ (u)ϕ′′ (u) + 8ce2ϕ(u)ϕ′ (u) = 0. (2.33)

Next, we solve the above equation. First, multiply it by e2ϕ/3/3 and then integrate, to

get that

ϕ′′ (u) = be2ϕ(u)/3 − ce2ϕ(u), (2.34)

where b ∈ R is a constant.

From (2.32), one obtains b = α8/3, which shows that b > 0. We can see that b has to

be positive also using the hypothesis c−K > 0. More precisely, from (1.6) we have that

K (u) = −be−8ϕ(u)/3 + c and, since c−K > 0 at any point, one gets that the constant

b has to be positive. Since K ′ (u) = 8beϕ(u)/3/3 > 0 and b > 0, it follows that ϕ′ > 0.

Moreover, multiplying by 2ϕ′ and then integrating equation (2.34), one obtains(ϕ′)2

= −3be−2ϕ/3 − ce2ϕ + a,

where a ∈ R.As ϕ′ > 0, we have

u =

∫ ϕ

ϕ0

dτ√−3be−2τ/3 − ce2τ + a

+ u0,

where ϕ is in some open interval I, ϕ0 ∈ I and a, b, u0 ∈ R are constants, b > 0. We

note that if c > 0, then a > 0.

Denote ϕ by ρ, u by u and v by v, we come to the conclusion.

To prove the last implication, (iv) ⇒ (iii), we rst note that equation (2.30) involve,

as we have already seen, that

ρ′′ = be−2ρ/3− ce2ρ, (2.35)

where b > 0 is a real constant. Consider the change of coordinates (u, v) =(b−3/8u, b−3/8v

)and rewrite the metric g as

g = e2(ρ(u)−(3/8) log b)(du2 + dv2

),

where ρ(u) = ρ (u (u)). Denote ϕ (u) = ρ(u)− (3 log b)/8 and by a direct computation,

one obtains that (2.35) is equivalent to

ϕ′′ (u) = e−2ϕ(u)/3 − ce2ϕ(u),

an equation that was already solved when we proved (ii) ⇒ (iii). Therefore, we come

to the conclusion.


Remark 2.23. If condition (ii) is satised, then K has to satisfy

3K ′′(c−K) + 3(K ′)2 + 8K(c−K)5/4 = 0

and, if c > 0, then(M2, (c−K)3/4g

)is a at surface and, trivially, a Ricci surface with

respect to c.

Remark 2.24. We have the following properties of the solutions of (2.30):

(i) the parameter b in the expression of the solution of (2.30) is not essential (and so

only the parameter a counts). Thus, we have a one-parameter family of solutions;

(ii) if ρ is a solution of (2.30), for some c, then ρ + α, where α is a real constant, is

also a solution of (2.30) for ce2α;

(iii) when c = 0, we note that if ρ is a solution of (2.30), then also ρ + constant is

a solution of the same equation, i.e, condition (i) from Theorem 2.22 is invariant

under the homothetic tranformations of the metric g. Then, we see that equation

(2.30) is invariant under ane changes of parameter u = αu + β, where α > 0.

Therefore, we solve equation (2.30) up to this change of parameter and an additive

constant of the solution ρ. The additive constant is the parameter that counts.

Chapter 3Complete biconservative

surfaces in R3 and S3

In this chapter, we extend the local classication results for biconservative surfaces in

N3(c), with c = 0 and c = 1, to global results, i.e., we construct complete biconservative

surfaces, with grad f = 0 at any point of on an open dense subset. Also, we study the

uniqueness of such surfaces in R3.

Most of the results presented here are original and they are also presented in [54],

[56], and [57]. Moreover, the results in Subsection 3.1.1 are presented here for the rst

time.

3.1 Complete biconservative surfaces in R3

In this section we construct, from extrinsic point of view, complete biconservative sur-

faces in R3 with grad f = 0 at any point of an open dense subset, and, from intrinsic

point of view, we construct a complete abstract surface(M2, g

)with K < 0 everywhere

and gradK = 0 at any point of an open dense subset ofM , that admits a biconservative

immersion in R3, dened on the whole M , with grad f = 0 on the open dense subset.

First, we recall a local extrinsic result which provides a characterization of biconser-

vative surfaces in R3.

Theorem 3.1 ([33]). Let M2 be a surface in R3 with (grad f)(p) = 0 for any p ∈ M .

Then, M is biconservative if and only if, locally, it is a surface of revolution, and the

curvature κ = κ(u) of the prole curve σ = σ(u), |σ′(u)| = 1, is a positive solution of

the following ODE

κ′′κ =7

4

(κ′)2 − 4κ4.

49

50 Chapter 3. Complete biconservative surfaces in R3 and S3

In [15] there was found the local explicit parametric equation of a biconservative

surface in R3.

Theorem 3.2 ([15]). Let M2 be a biconservative surface in R3 with (grad f)(p) = 0

for any p ∈M . Then, locally, the surface can be parametrized by

XC0(ρ, v) =

(ρ cos v, ρ sin v, uC0

(ρ)),

where

uC0(ρ) =

3

2C0

(ρ1/3

√C0ρ2/3 − 1 +

1√C0

log

(√C0ρ

1/3 +

√C0ρ2/3 − 1

))

with C0 a positive constant and ρ ∈(C

−3/20 ,∞

).

We denote by SC0the image XC0

((C

−3/20 ,∞

)× R

). We note that any two such

surfaces are not locally isometric, so we have a one-parameter family of biconservative

surfaces in R3. These surfaces are not complete.

We dene the boundary of SC0by SC0

\ SC0, where SC0

is the closure of SC0in

R3.

The boundary of SC0is the circle

(C

−3/20 cos v, C

−3/20 sin v, 0

), which lies in the

Oxy plane. At a boundary point, the tangent plane to SC0is parallel to Oz. Moreover,

along the boundary, the mean curvature function is constant fC0=(2C

3/20

)/3 and

grad fC0= 0.

Proposition 3.3. Let SC0and SC′

0. Assume that we can glue them along a curve at the

level of C∞ smoothness. Then SC0and SC′

0coincide or one of them is the symmetric

of the another with respect to the plane where the common boundary lies.

Proof. We consider SC0and SC′

0determined by

XC0(ρ, v) = ρ cos v e1 + ρ sin v e2 + uC0

(ρ) e3,

and

XC′0(ρ, v) = (ρ cos v + a1) f1 + (ρ sin v + a1) f2 +

(uC′

0(ρ) + a3

)f3,

where uC0(ρ), uC′

0(ρ) are given in Theorem 3.2, e1, e2, e3 is the canonical basis in R3,

f1, f2, f3is a positively oriented orthonormal basis of R3 and a1, a2, a3 ∈ R. Assume

that we can glue SC0and SC′

0along a curve γ = γ(s), γ′(s) = 0, for any s, at the level

of C∞ smoothness. In this case we have

γ(s) ∈ SC0∩ SC′

0

ηC0(γ(s)) || ηC′

0(γ(s))

HC0(γ(s)) = HC′

0(γ(s))(

grad∣∣∣HC0

∣∣∣ ) (γ(s)) =

(grad

∣∣∣HC′0

∣∣∣ ) (γ(s))

, (3.1)

3.1. Complete biconservative surfaces in R3 51

for any s, where the mean curvature vector eld HC0is given by HC0

= fC0ηC0

/2. For

SC0we have

ηC0(ρ, v) =

XC0,ρ×XC0,v∣∣∣XC0,ρ×XC0,v

∣∣∣= − 1√

C0ρ1/3cos v e1 −

1√C0ρ1/3

sin v e2 +

√C0ρ2/3 − 1

C0ρ2/3e3

and the mean curvature function

fC0(ρ, v) =

(1 +

(u′C0(ρ))2)−3/2

u′′C0(ρ) +

u′C0(ρ)

(1 +

(u′C0(ρ))2)

ρ

=

2

3√C0ρ4/3

> 0.

It follows that fC0(ρ, v) = fC0

(ρ), fC0= 2

∣∣∣HC0

∣∣∣, and(grad fC0

)(ρ, v) =

1

1 +(u′C0(ρ))2 f ′C0

(ρ) XC0,ρ(ρ, v)

= − 8

9C3/20 ρ3

((C0ρ

2/3 − 1)cos v e1 +

(C0ρ

2/3 − 1)sin v e2+

+

√C0ρ2/3 − 1 e3

).

Similar formulas hold for SC′0. Now, let us consider

(ρ1(s), v1(s)) =(X−1C0

γ)(s) and (ρ2(s), v2(s)) =

(X−1C′

0

γ)(s).

We can rewrite (3.1) as

XC0(ρ1(s), v1(s)) = XC′

0(ρ2(s), v2(s))

ηC1 (ρ1(s), v1(s)) = ηC′0(ρ2(s), v2(s))

fC0(ρ1(s), v1(s)) = fC′

0(ρ2(s), v2(s))(

grad fC0

)(ρ1(s), v1(s)) = (grad fC′

0) (ρ2(s), v2(s))

, (3.2)

for any s, where ρ1(s) ≥ C−3/20 and ρ2(s) ≥

(C ′0

)−3/2.

First, we can notice that C0ρ2/31 (s)− 1 = 0 if and only if C ′

0ρ2/32 (s)− 1 = 0. Next,

we consider two cases.

In the rst case, when C0ρ2/31 (s)−1 = 0 for any s, by a straightforward computation,

from the third relation of (3.2), we can see that C0 = C ′0 and ρ1(s) = ρ2(s) = C

−3/20 ,


for any s. Moreover, uC0(ρ1(s)) = 0 and uC′

0(ρ2(s)) = 0. Then, from the rst relation

we get a1 = a2 = a3 = 0 and ⟨e1, f3⟩ = ⟨e2, f3⟩ = 0, i.e., e3 = ±f3. Therefore, SC0

and SC′0coincide or one of them is the symmetric of another with respect to the ane

plane where the common boundary lies.

In the second case, we suppose that there exists s0 such that C0ρ2/31 (s0) − 1 = 0.

It follows that also C ′0ρ

2/32 (s0) − 1 = 0. Thus, we get that C0ρ

2/31 (s) − 1 > 0 and

C ′0ρ

2/32 (s) − 1 > 0 around s0. By direct computation, from (3.2), we obtain C0 = C ′

0,

a1 = a2 = a3 = 0, ρ1(s) = ρ2(s) around s0, and ⟨e3, f3⟩ = 1, i.e., e3 = f3. Therefore,

in this case SC0and SC′

0coincide.

However, we must then check that we have a smooth gluing.

Theorem 3.4. If φ : M2 → R3 is a biconservative surface with grad f = 0 at any

point, then there exists a unique C0 such that φ(M) ⊂ SC0.

Proof. From Theorem 3.2, it is easy to see that any point of M admits an open neigh-

borhood which is an open subset of some SC0. Let us consider p0 ∈ M . Then, using

Proposition 3.3, it follows that there exists a unique C0 such that φ(U) ⊂ SC0, where

U is an open neighborhood of p0. If V denotes the set of all points of M such that

they admit open neighborhoods which are open subsets of that SC0, then the set V is

non-empty, open and closed in M . Indeed, it is clear that V is non-empty and open. In

order to prove that V is closed, we x a point q0 ∈ V and note that, from Theorem 3.2,

there exists an open neighborhood W of q0 such that W is an open subset of some SC′0.

If C0 = C ′0, we obtain that q0 ∈ V , so the set V is closed. If we assume that C0 = C ′

0,

since W ∩ V is non-empty, open and W ∩ V ⊂ SC0∩ SC′

0, using again Proposition 3.3,

it follows that SC0= SC′

0and V is a closed set.

Thus, as M is connected, it follows that V =M .

In order to obtain a complete biconservative surface in R3, we can expect to glue

along the boundary two biconservative surfaces of type SC0corresponding to the same

C0 (the two constants have to be the same) and symmetric to each other, at the level

of C∞ smoothness.

We have the following global extrinsic result.

Theorem 3.5 ([49, 54]). If we consider the symmetry of Graf uC0, with respect to the

Oρ(= Ox) axis, we get a smooth, complete, biconservative surface SC0in R3. Moreover,

its mean curvature function fC0is positive and grad fC0

is dierent from zero at any

point of an open dense subset of SC0.

Proof. Obviously, limρC

−3/20

uC0(ρ) = 0. As u′

C0(ρ) > 0 for any ρ ∈

(C

−3/20 ,∞

), we

can think ρ as a function of u and

XC0(u, v) =

(ρC0

(u) cos v, ρC0(u) sin v, u

), u ∈ (0,∞).


If we consider the symmetry of Graf uC0, when ρ ∈

(C

−3/20 ,∞

)with respect to the

Oρ = Ox axis, we get a smooth complete biconservative surface SC0in R3, given by

XC0(u, v) =

(xC0

(u) cos v, xC0(u) sin v, u

), (u, v) ∈ R,

where

xC0(u) =

ρC0

(u), u > 0

C−3/20 , u = 0

ρC0(−u), u < 0

is a smooth function. Moreover, the curvature function f is positive and grad f is

dierent from zero at any point of an open dense subset of SC0. We note that grad fC0

vanishes only along the boundary of SC0.

Remark 3.6. The prole curve σC0=(ρ, 0, uC0

(ρ))

≡(ρ, uC0

(ρ))can be repara-

metrized as

σC0(θ) =

(σ1C0(θ), σ2

C0(θ))

= C−3/20

((θ + 1)3/2, 32

(√θ2 + θ + log

(√θ +

√θ + 1

))), θ > 0,

(3.3)

and now XC0= XC0

(θ, v).

Remark 3.7. The boundary of SC0coincide with the boundary of SC0

as a subset of

SC0, i.e., the intersection between the closure of SC0

in SC0and the closure of SC0

\SC0

in SC0.

Proposition 3.8. The homothety of R3, (x, y, z) → C0(x, y, z), renders S1 onto SC−2/30

.

For the sake of completeness we represent in Figures 3.1, 3.2 and 3.3 the surfaces

SC0, SC0

, and the prole curve of SC0, respectively, when C0 = 91/3 (we will see that

this constant corresponds to the constant C0 = 1).

Now, we change the point of view and construct, from intrinsic point of view, com-

plete biconservative surfaces in R3 with grad f = 0 on an open and dense subset. Using

local intrinsic characterization (Theorem 2.22), for c = 0, one gets the next result.


)be a Riemannian surface with Gaussian curvature K

satisfying (gradK)(p) = 0 and K(p) < 0 at any point p ∈ M . Consider X1 =

gradK/| gradK| and X2 ∈ C(TM) be two vector elds on M such that X1(p), X2(p)is a positively oriented orthonormal basis at any point p ∈M . Then X2 (X1K) = 0 and

∇X2X2 = (3 (X1K) /(8K))X1 if and only if the Riemannian metric g can be locally

written as

gC0(u, v) = C0 (coshu)6 (du2 + dv2), u > 0,

where C0 ∈ R is a positive constant.


Figure 3.1: The surface SC0. Figure 3.2: The complete surface SC0

.

Figure 3.3: The prole curve of SC0.

Proof. For c = 0, equation (2.30) becomes

3ρ′′′(u) + 2ρ′(u)ρ′′(u) = 0, (3.4)

with initial conditions ρ′ > 0 and ρ′′ > 0. We note that since K = −e−2ρ(u)ρ′′(u) < 0,

it is clear that ρ′′(u) > 0 for any u.

By a straightforward computation, we get the unique solution of (3.4)

ρ(u) = a

∫ u

u′0

1− e−2a(τ+u0)/3

1 + e−2a(τ+u0)/3dτ + b1, u ∈ I, (3.5)

where a, b1, u0 ∈ R, I is an open interval and u′0 ∈ I is arbitrary xed.

Next, we can assume that K ′(u) > 0 and we compute the integral in (3.5). First,

we prove that K ′(u) > 0 if and only if u+ u0 > 0.


Since

K(u) = −e−2ρ(u)ρ′′(u), u ∈ I, (3.6)

we have that

K ′(u) = e−2ρ(u)(2ρ′(u)ρ′′(u)− ρ′′′(u)

)> 0, u ∈ I,

if and only if

2ρ′(u)ρ′′(u)− ρ′′′(u) > 0, u ∈ I. (3.7)

From (3.5) we get

ρ′′′(u) = −8a3e−2a(u+u0)/3

(1− e−2a(u+u0)/3

)9(1 + e−2a(u+u0)/3

)3 .

If we replace the rst, the second and the third derivatives of ρ in (3.7), we obtain

that K ′(u) > 0 if and only if a3(1− e−2a(u+u0)/3

)> 0. It is easy to check that this is

equivalent to u+ u0 > 0 if either a > 0 or a < 0.

Therefore, the solution is

ρ(u) = a

∫ u

u′0

1− e−2a(τ+u0)/3

1 + e−2a(τ+u0)/3dτ + b1, u ∈ I, u+ u0 > 0,

where b1, u0 ∈ R, a ∈ R∗, I is an open interval and u′0 ∈ I is arbitrary xed.

Then, we denote by I the integral in (3.5) and, in order to compute it, we consider

some changes of variables. First, if we denote by s = −2a (τ + u0) /3, we obtain

I =(u− u′0

)+

3

a

∫ −2a(u+u′0)/3

−2a(u0+u′0)/3

es

1 + esds.

We continue with an other substitution t = es and one gets

I =(u− u′0

)+

3

a

(log(1 + e−2a(u+u′0)/3

)− log

(1 + e−2a(u0+u′0)/3

)).

If we substitute this expression of I in (3.5), it follows that

ρ(u) = 3 log(1 + e−2a(u+u′0)/3

)+ au+ b2, u ∈ I, u+ u0 > 0,

where b2, u0 ∈ R, a ∈ R∗.

Now, we consider two cases: if a > 0 and if a < 0. In the rst case, we make the

change of coordinates (u, v) = (3u/a− u0, 3v/a) and in the second one, we consider the

change of coordinates (u, v) = (−3u/a− u0,−3v/a). We note that in both situations,

one gets

ρ (u) = ρ (u (u)) = 3 log (cosh u) + b,


where b ∈ R, and since g =(9/a2

)e2ρ(u)

(du2 + dv2

), we nd

gC0 = C0 (cosh u)6 (du2 + dv2

),

where (W ; u, v) is an isothermal chart positively oriented, u > 0, and C0 ∈ R is a

positive constant.

Remark 3.10. We note that, when c = 0, we have a one-parameter family of solutions

of equation (2.30), i.e., gC0 = C0(coshu)6(du2 + dv2

), C0 being a positive constant.

Concerning the complete biconservative surfaces in R3, with grad f = 0 at any point

of an open dense subset, we have the next global intrinsic result.

Theorem 3.11. Let(R2, gC0 = C0 (coshu)

6 (du2 + dv2))

be a surface, where C0 ∈ Ris a positive constant. Then we have:

(i) the metric on R2 is complete;

(ii) the Gaussian curvature is given by

KC0(u, v) = KC0(u) = − 3

C0 (coshu)8 < 0, K ′

C0(u) =

24 sinhu

C0 (coshu)9 ,

and therefore gradKC0 = 0 at any point of R2 \Ov;

(iii) the immersion φC0 :(R2, gC0

)→ R3 given by

φC0(u, v) =(σ1C0

(u) cos(3v), σ1C0(u) sin(3v), σ2C0

(u))

is biconservative in R3, where

σ1C0(u) =

√C0

3(coshu)3 , σ2C0

(u) =

√C0

2

(1

2sinh(2u) + u

), u ∈ R.

Proof. In order to prove (i), we use Proposition 1.8.

Consider g0 = du2 + dv2 the Euclidean metric on R2, which is complete. Then,

denote by g the Riemannian metric g = (coshu)6g0, and note that

g − g0 =((coshu)6 − 1

)g0

is non-negative denite at any point of R2. Therefore g is also complete and since

gC0 = C0g, it follows that(R2, gC0

)is complete.

To prove (ii), we consider the formula (3.6), with φ(u) = log(√

C0 (coshu)3)and

obtain that the Gaussian curvature KC0(u, v) is equal to

KC0(u, v) = KC0(u) = − 3

C0 (coshu)8


and

K ′C0(u) =

24

C0

sinhu

(coshu)9.

Therefore, K ′C0(u) > 0 if and only if u > 0, K ′

C0(u) < 0 if and only if u < 0, and

K ′C0(0) = 0. Since

(gradKC0) (u, v) =1

C0e−6 log(coshu)K ′

C0(u)∂u,

we have gradKC0 = 0 at any point of R2 \Ov, which is an open dense subset of R2.

We begin the proof of (iii), recalling that, from Remark 3.6, we have the result

which says that if we consider a biconservative surface in R3, with non-constant mean

curvature, then, locally, it is a surface of revolution with the prole curve

σ+C0(θ) =

(σ1C0(θ), σ2

C0(θ))

= C−3/20

((θ + 1)3/2,

3

2

[√θ2 + θ + log(

√θ +

√θ + 1)

]), θ > 0,

and which admits the local parametrization

X+C0(θ, v) = C

−3/20

((θ + 1)3/2 cos v, (θ + 1)3/2 sin v,

3

2

[√θ2 + θ + log(

√θ +

√θ + 1)

]), θ > 0, v ∈ R.

To compute the metric on this surface, we rst need the coecients of the rst

fundamental form

E+C0(θ, v) =

1

C30

9(θ + 1)2

4θ, F+

C0(θ, v) = 0, G+

C0(θ, v) =

1

C30

(θ + 1)3.

Thus, the Riemannian metric on this surface is

g+C0(θ, v) =

1

C30

(9(θ + 1)2

4θdθ2 + (θ + 1)3dv2

).

If we consider the change of coordinates (θ, v) =((sinhu)2 , 3v

), where u = 0, ones

obtains

g+C0(u, v) =

9

C30

(coshu)6(du2 + dv2

).

Since C0 is an arbitrary positive constant, we can consider C0 = (9/C0)1/3, where C0

is the positive constant corresponding to gC0 , and therefore g+C0

= gC0 .

Then, we dene φC0 as: for u > 0, φC0(u, v) is obtained by rotating the prole

curve

σ+(9

C0

)1/3(u) =

(σ1(

9C0

)1/3 (u) , σ2(

9C0

)1/3 (u)

)

=

√C0

3

((coshu)

3,3

2(sinhu coshu+ log (sinhu+ coshu))

)=

√C0

3

((coshu)

3,3

2

(1

2sinh 2u+ u

)), u > 0,


and for u < 0, φC0(u, v) is obtained by rotating the prole curve

σ−(9

C0

)1/3(u) =

(σ1(

9C0

)1/3 (−u) ,−σ2(9

C0

)1/3 (−u)

)

=C

1/20

3

((coshu)

3,3

2

(1

2sinh 2u+ u

)), u < 0.

Now, it is easy to see that we have a biconservative immersion, in fact a biconser-

vative embedding from the whole(R2, gC0

)in R3, given by

X(9C0

)1/3(u, v) =

√C0

3

((coshu)3 cos 3v, (coshu)3 sin 3v,

3

2

(1

2sinh 2u+ u

)).

By simple transformations of the metric,(R2, gC0

)becomes a Ricci surface or a

surface with constant Gaussian curvature.

Theorem 3.12. Consider the surface(R2, gC0

). Then

(R2,

√−KC0gC0

)is complete,

satises the Ricci condition and can be minimally immersed in R3 as a helicoid or a

catenoid.

Proposition 3.13. Consider the surface(R2, gC0

). Then

(R2,−KC0gC0

)has constant

Gaussian curvature 1/3 and it is not complete. Moreover,(R2,−KC0gC0

)is the uni-

versal cover of the surface of revolution in R3 given by

Z(u, v) =

(α(u) cosh

(√3

av

), α(u) sinh

(√3

av

), β(u)

), (u, v) ∈ R2,

where a ∈ (0,√3] and

α(u) =a

coshu, β(u) =

∫ u

0

√(3− a2) cosh2 τ + a2

cosh2 τdτ .

Remark 3.14. When a =√3, the immersion Z has only umbilical points and the

image Z(R2)is the round sphere of radius

√3, without the North and the South poles.

Moreover, if a ∈ (0,√3), then Z has no umbilical points.

Concerning the biharmonic surfaces in R3 we have the following non-existence result.

Theorem 3.15 ([19, 20]). There exists no proper-biharmonic surface in R3.


3.1.1 Uniqueness of complete biconservative sur-

faces in R3

In this subsection, we give some uniqueness results concerning Theorem 3.11, under

some additional assumptions.

Theorem 3.16. Let φ :M2 → R3 be a non-CMC surface. Assume that

W = p ∈M | (grad f)(p) = 0

has only one connected component, M \W has non-empty interior and the boundaries,

in M , of Int(M \W ) and M \W coincide, i.e., ∂M Int(M \W ) = ∂M (M \W ). Then

M cannot be biconservative.

Proof. Let us consider an arbitrary point p0 ∈ ∂MW = ∂M (M \W ) = ∂M Int(M \W ).

Since p0 ∈M , it follows that there exists an open subset U0 ofM , such that p0 ∈ U0 and

φ|U0: U0 → R3 is an embedding. Thus, we can identify U0 with its image φ (U0) ⊂ R3

and then U0 can be seen as a regular surface in R3.

We note that p0 ∈ ∂MW∩U0 leads to the existence of a sequence(p1n)n∈N∗ ⊂W∩U0,

p1n = p0, for any n ∈ N∗, which converges to p0, with respect to the intrinsic distance

function dM on M , and, similarly, from p0 ∈ ∂M Int(M \W ) ∩ U0 it follows that there

exists a sequence(p2n)n∈N∗ ⊂ Int(M \W )∩U0, p

2n = p0, for any n ∈ N∗, which converges

to p0, with respect to dM . It is clear that we can identify p1n = φ(p1n)and p2n = φ

(p2n).

Now, since W is connected and grad f = 0 at any point of W , from Theorem 3.4,

one obtains that there exists a unique C0 such that φ(W ) is an open set in SC0. Then,

as W ∩ U0 is open in W , it is clear that φ (W ∩ U0) is open in SC0. In fact, using the

identication φ (W ∩ U0) = W ∩ U0, we have W ∩ U0 open in SC0. We recall that SC0

is open in the complete surface SC0, and then W ∩ U0 is also open in SC0

.

Further, we denote by d0 the distance function on R3 and by dSC0

the intrinsic

distance function on SC0. Obviously, the convergence of the sequence

(p1n)to p0, with

respect to the distance dM , implies the convergence of(p1n)to p0, with respect to the

distance d0.

The sequence(p1n)was chosen in W ∩ U0, so

(p1n)⊂ SC0

. As SC0is a closed set in

R3, we obtain that p0 ∈ SC0. Therefore,

(p1n)converges to p0 also with respect to the

distance dSC0

.

We have already seen that W ∩ U0 is open in U0 and also in SC0. Then, the mean

curvature functions f and fC0corresponding to M and SC0

, respectively, coincide on


W ∩ U0. Therefore, for any n ∈ N∗ one has

f(p1n)= fC0

(p1n)

(grad f)(p1n)=(grad fC0

) (p1n)∣∣(grad f) (p1n)∣∣ = ∣∣∣(grad fC0

) (p1n)∣∣∣

(∆f)(p1n)=(∆fC0

) (p1n) . (3.8)

From the convergence of(p1n)to the same p0, with respect to the both distance functions

dM and dSC0

, and from the third equation in (3.8), one gets

|(grad f) (p0)| =∣∣∣(grad fC0

)(p0)

∣∣∣ .As p0 ∈ ∂MW , we have (grad f) (p0) = 0 and then

(grad fC0

)(p0) = 0, that means p0

belongs to the boundary, in R3, of SC0. Thus, fC0

(p0) = 2C3/20 /3 = 0.

Using the rst equation in (3.8) and the convergence of(p1n)to p0, with respect to

dM and dSC0

, we obtain f (p0) = fC0(p0).

Assume that M is biconservative. From (1.19) applied for c = 0, one has

f(p1n)(∆f)

(p1n)+∣∣(grad f) (p1n)∣∣2 − f4

(p1n)= 0,

for any n ∈ N∗. We may pass to the limit with respect to the distance dM in the above

equation and obtain

f (p0) (∆f) (p0) + |(grad f) (p0)|2 − f4 (p0) = 0.

According to the above observations, this is equivalent to

fC0(p0) (∆f) (p0)− f4

C0(p0) = 0. (3.9)

We also have seen that there exists a sequence(p2n)n∈N∗ ⊂ Int(M \ W ) ∩ U0 which

converges to p0, with respect to the distance dM . Since grad f = 0 at any point of

Int(M \W )∩U0 and Int(M \W )∩U0 is open inM , it is easy to see that (grad f)(p2n)= 0

and (∆f)(p2n)= 0, for any n ∈ N∗. Passing to the limit with respect to the distance

dM in the above two relations we get (grad f) (p0) = 0 and (∆f) (p0) = 0.

Substituting (∆f) (p0) = 0 in (3.9), one obtains fC0(p0) = 0. Therefore we have a

contradiction.

Theorem 3.17. Let φ :M2 → R3 be a biconservative surface. Assume that

W = p ∈M | (grad f)(p) = 0


is dense and it has two connected components, W1 and W2. Assume that the boundaries

of W1 and W2 in W coincide and their common boundary is a smooth curve in M .

Then, there exists a unique C0 such that φ(M) ⊂ SC0. Moreover, if M is complete

and simply connected, then up to isometries of the domain and codomain, φ is the map

given in Theorem 3.11.

Proof. Let us consider p0 ∈ ∂MW1 = ∂MW2. There exists an open set U0 in M ,

such that p0 ∈ U0 and φ|U0: U0 → R3 is an embedding. Thus, we can identify

U0 = φ (U0) ⊂ R3.

SinceW1 andW2 are connected and grad f = 0 at any point of them, from Theorem

3.4, one obtains that there exist C0 and C ′0 such that φ (W1) is an open subset of SC0

and φ (W2) is an open subset of SC′0.

It is clear that U0 ∩W1 is open in W1 and then φ (U0 ∩W1) = U0 ∩W1 is open in

SC0, and analogous, U0 ∩W2 is open in W2 and then φ (U0 ∩W2) = U0 ∩W2 is open in

SC′0.

We note that, as W is dense in M , one has M = W ∪ ∂MW = W1 ∪W2 ∪ ∂MW1.

Therefore,

U0 = U0 ∩M

= (U0 ∩W1) ∪ (U0 ∩W2) ∪(U0 ∩ ∂MW1

).

We consider U0 ∩ ∂MW1 as the image of γ : I → U0, γ′(s) = 0, for any s ∈ I. It is

clear that γ(s) ∈ SC0and

(grad fC0

)(γ(s)) = (grad f)(γ(s)) = 0, i.e., γ(s) belongs to

the boundary of SC0, for any s ∈ I, which is a circle of radius C

−3/20 . With the same

argument, γ(s) belongs to the boundary of SC′0, for any s ∈ I, which is a circle of radius

C′−3/20 . Therefore, C0 = C ′

0 and φ(M) ⊂ SC0.

If M is complete, φ : M → SC0is a covering space with the projection φ and thus

φ(M) = SC0. Moreover, if M is also simply connected, then M is a universal covering

of SC0with the projection φ.

The map φC0 :(R2, gC0

)→ SC0

given in Theorem 3.11 is also a universal covering

projection and therefore there exists an isometry Θ between (M, g) and(R2, gC0

)such

that φC0 Θ = φ.

In the following, we restrict ourself to the case when φ :M2 → R3 is an embedding,

i.e., S = φ(M) is a regular surface in R3. We start with the next result.

Theorem 3.18. Let S a complete biconservative regular surface in R3. Denote by

W = p ∈ S | (grad f)(p) = 0


and assume that W is non-empty and W0 is a connected component of W . Then, there

exists a unique C0 > 0 such that W0 = SC0. Moreover, the closure of W0 in S coincides

with the closure of SC0in SC0

.

Proof. We note that since W0 is a connected component of W and W is an open subset

of S, then W0 is closed and also open in W . We also have that W0 is a maximal

connected subset of W with respect to the inclusion. It is clear that W0 is also open in

S.

We denote by ∂SW0 the boundary of W0 in S and prove that (grad f) (q) = 0, for

any q ∈ ∂SW0. In order to show this, we assume that there exists a point q0 ∈ ∂SW0

such that (grad f) (q0) = 0. Then, it follows that one has an open ball B2 (q0; r0) in S,

r0 > 0, such that grad f is dierent from zero at any point of it.

Obviously, q0 belongs to the closure ofW0 in S, and then B2 (q0; r0)∩W0 = ∅. SinceB2 (q0; r0) and W0 are connected sets, we get that B2 (q0; r0) ∪W0 is also connected.

Moreover, as grad f is dierent from zero at any point of B2 (q0; r0) it follows that

B2 (q0; r0) ⊂W and, therefore B2 (q0; r0) ∪W0 is a connected subset of W . Now, from

the maximality of W0 in W , we have B2 (q0; r0) ∪ W0 = W0, i.e., B2 (q0; r0) ⊂ W0.

Clearly, we obtain that q0 ∈W0, and this is false because q0 ∈ ∂SW0 and W0 is open S.

Thus, one has (grad f) (q) = 0, for any q ∈ ∂SW0.

It is easy to note that since W0 is connected and grad f = 0 at any point of W0,

from Theorem 3.4, one obtains that there exists a unique C0 such that W0 is open in

SC0. Moreover, we will prove that, in this case, W0 = SC0

.

Let us consider σC0: (0,∞) → R2 the prole curve of SC0

, σC0(0,∞) ⊂ SC0

. We can

reparametrize σC0by arc-length, such that the new curve, denoted also by σC0

= σC0(θ),

has the same orientation as the initial one, is dened on (0,∞) and in zero has the same

limit point(C

−3/20 , 0

)on the boundary of SC0

. The new curve σC0is a parametrized

geodesic of SC0and we recall that

(grad fC0

)(σC0

(θ))= 0, for any θ > 0.

Next, we will prove that σC0(0,∞) ⊂ W0. Clearly, there exists a point θ0 ∈ (0,∞)

such that σC0(θ0) ∈ W0. Since σC0

is continuous and W0 is open in SC0, it follows

that exists ε0 > 0 such that (θ0 − ε0, θ0 + ε0) ⊂ (0,∞) and σC0(θ0 − ε0, θ0 + ε0) ⊂W0.

Assume that σC0(0,∞) ⊂W0, i.e., there exists θ

′ ∈ (0,∞) \ (θ0 − ε0, θ0 + ε0) such that

σC0(θ′) ∈W0.

Assume that θ′ ≥ θ0 + ε0. Denote

Ω =θ | θ > θ0, σC0

(θ) ∈W0

and θ1 = inf Ω.

We note that θ1 ≥ θ0 + ε0. Indeed, if we assume that θ1 < θ0 + ε0, since θ1 = inf Ω,

it follows that there exists θ2 ∈ Ω such that θ1 ≤ θ2 < θ0 + ε0. It is easy to see that


θ2 ∈ (θ0, θ0 + ε0) ⊂ (θ0 − ε0, θ0 + ε0), thus σC0(θ2) ∈ W0. But, this is a contradiction

because θ2 ∈ Ω, and then σC0(θ2) ∈W0.

Next, we show that σC0(θ1) ∈ W0. Indeed, if σC0

(θ1) ∈ W0, it follows that there

exists ε1 > 0 such that (θ1 − ε1, θ1 + ε1) ⊂ (0,∞) and σC0(θ1 − ε1, θ1 + ε1) ⊂ W0.

Therefore, we obtain a contradiction because that implies the non-existence of a se-

quence from Ω which converges to θ1, and so θ1 cannot be an inmum of Ω.

Moreover, we have σC0(θ) ∈ W0 for any θ ∈ [θ0, θ1). We have already seen that

σC0(θ0) ∈W0. Assume that there exists θ ∈ (θ0, θ1) such that σC0

(θ)∈W0. It follows

that θ ∈ Ω. But θ < θ1, so we obtain a contradiction with the fact that θ1 is an inmum.

Since σC0(θ) ∈W0 for any θ ∈ [θ0, θ1), it is clear that σC0

(θ1) belongs to the closure

of W0 in SC0, denoted by W0

SC0 . As σC0(θ1) ∈W0 and W0 is open in SC0

, one obtains

that σC0(θ1) ∈ ∂

SC0W0, i.e. σC0(θ1) belongs to the boundary of W0 in SC0

.

We have seen that σC0(θ) ∈W0, for any θ ∈ (θ0 − ε0, θ1). Now, sinceW0 is open in S

and σC0is a parametrized geodesic of SC0

, we get that σC0dened of (θ0 − ε0, θ1) is also

a parametrized geodesic of S. As S is complete, we can consider a parametrized geodesic

σS dened on whole R, such that σS∣∣(θ0−ε0,θ1) = σC0

∣∣∣(θ0−ε0,θ1)

. It is clear that, since

σS and σC0are continuous on R and on (0,∞), respectively, that σS (θ1) = σC0

(θ1).

Now, we note that σS (θ1) ∈ ∂SW0, (σS (θ1) ∈W0

Sand σS (θ1) = σC0

(θ1) ∈W0).

Therefore (grad f)(σS (θ1)

)= (grad f)

(σC0

(θ1))= 0.

Further, we use the fact that sinceW0 is an open set in both S and SC0, then we have

equality between the mean curvature functions of S and of SC0at every point of W0,

i.e., f |W0= fC0

∣∣∣W0

, and between their gradients, i.e., (grad f)|W0=(grad fC0

)∣∣∣W0

.

Clearly,

(grad f)(σS(θ)

)=(grad fC0

)(σC0

(θ)), θ ∈ (θ0 − ε0, θ1) ,

and, then, ∣∣(grad f) (σS(θ))∣∣ = ∣∣∣(grad fC0

)(σC0

(θ))∣∣∣ , θ ∈ (θ0 − ε0, θ1) .

We may pass to the limit in the above equation and obtain∣∣(grad f) (σS (θ1))∣∣ = ∣∣∣(grad fC0

)(σC0

(θ1))∣∣∣ .

Therefore, one gets a contradiction, because we have already seen that

(grad f)(σS (θ1)

)= (grad f)

(σC0

(θ1))= 0,

and(grad fC0

)(σC0

(θ1))= 0, since σC0

(θ1) ∈ SC0.

Thus, σC0(θ) ∈W0, for any θ ≥ θ0 + ε0.


In the same way, we can prove that σC0(θ) ∈W0, also for any θ ≤ θ0 − ε0.

Finally, we obtain that σC0(0,∞) ⊂W0.

Now, we recall that SC0is open in SC0

, and then W0 is also open in S ∩ SC0. Since

SC0is complete, we can consider a parametrized geodesic σC0

dened on whole R, suchthat σC0

∣∣∣(0,∞)

= σC0= σS

∣∣(0,∞)

. Obviously, σS(0) = σC0(0), thus σC0

∣∣∣[0,∞)

= σS∣∣[0,∞)

and the closure of W0 in S coincides with the closure of SC0in SC0

.

Further, we consider γ the curve parametrized by arc-length, dened on the whole

R, which gives the boundary of SC0in SC0

. Clearly, γ is a parametrized geodesic in

SC0. According to the above observations, it follows that there exist a, b ∈ R with

a < b, such that γ(a, b) belongs also to S. Moreover, γ dened on (a, b) is also geodesic

in S because, along it, the normal vector eld to S coincide with the normal vector

eld to SC0and it is collinear with the principal unit normal vector of γ|(a,b). Since

S is complete, we can consider γS : R → S a parametrized geodesic on S such that

γS∣∣(a,b)

= γ|(a,b).We note that the maximal interval which contains (a, b) and has the property that

its image by γ is contained in S is R. Indeed, assume that there exists b′, b ≤ b′ < ∞,

such that γ(a, b′) ⊂ S and γ(b′) /∈ S. It is now easy to see that, as γS∣∣(a,b′)

= γ|(a,b′),γ(b′) = γS(b′) ∈ S, and thus we get a contradiction.

Therefore, W0 = SC0.

Remark 3.19. The proof of Theorem 3.18 can be summarizing in Figure 3.4, where

the yellow region represents the connected componentW0 of W in S, and the surface of

revolution represented with the color green is the corresponding SC0(given by Theorem

3.4). It is suggested that, in fact, all the meridians of SC0which intersect W0 are

contained in W0 and then, as the boundary of W0 in S has to be the whole circle which

gives the boundary of SC0in SC0

, W0 = SC0.

A rst consequence of the above result is the following theorem.

Theorem 3.20. Let S be a biconservative regular surface in R3. Assume that S is

compact. Then S is CMC, and therefore a round sphere.

Proof. Assume that S is non-CMC, i.e., W = p ∈ S | (grad f)(p) = 0 is a non-empty

subset of S. Let us consider W0 a connected component of W . From Theorem 3.18,

it follows that there exists C0 > 0 such that W0 = SC0and the closure of W0 in S

coincides with the closure of SC0in SC0

. Since SC0is unbounded in R3, we obtain that

S is unbounded and this is a contradiction with the compactness of S.

The last part follows from the famous theorem of Alexandrov (see, for example

[52]).

Other consequence of Theorem 3.18 is the next result.


Figure 3.4: The idea of the proof of Theorem 3.18.

Theorem 3.21. Let S be a complete regular surface in R3. Assume that

W = p ∈ S | (grad f)(p) = 0

is non-empty and is connected. Then S cannot be biconservative.

Proof. Assume that S is biconservative. From Theorem 3.18 it follows that there exists

C0 > 0 such that W = SC0and the closure of W in S coincides with the closure of SC0

in SC0.

It is easy to see that W is not dense in S, because if we assume that it is, then S =

WS= S

SC0

C0. This means that S is a surface with boundary and this is a contradiction


with the regularity of S. Therefore, S \WSis non-empty and open in S. We note that

∂SW = ∂S(S \W )

= S \WS ∩WS

= (S \W ) ∩WS

=(S \ Int

(W

S))

∩WS

= S \WSS

∩WS

= ∂S(S \WS

)and ∂SW is a circle of radius C

−3/20 .

Further, let us consider p0 ∈ ∂SW = ∂S(S \WS

). Then there exists a sequence(

p1n)n∈N∗ in W , with p1n = p0, for any n ∈ N∗ which converges to p0, with respect to the

distance function dS on S, and another sequence(p2n)n∈N∗ in S \WS

, with p2n = p0, for

any n ∈ N∗ which converges to p0, with respect to the same dS . Since W and S \WS

are open in S, grad f is dierent from zero at any point of W and grad f vanishes at

any point of S \WS, we can use the same argument as in the proof of Theorem 3.16,

to obtain a contradiction.

Therefore, our assumption is false, and S is not biconservative.

Theorem 3.22. Let S be a complete biconservative regular surface in R3. If S is

non-CMC, then S = SC0.

Proof. Since S is non-CMC, then

W = p ∈ S | (grad f)(p) = 0

is non-empty. Let us consider W0 a connected component of W . Then there exists a

unique C0 such that W0 = SC0and W0

S= SC0

SC0 . We denote, the surface SC0by S+

C0.

Let p0 ∈ ∂SW0, i.e., p0 is a point on the circle of radius C−3/20 . We have three cases.

First, assume that there exists ε0 > 0 such that grad f vanishes at any point of

B2 (p0; ε0) \(B2 (p0; ε0) ∩W0

S). Then there exists a sequence

(p1n)n∈N∗ in W0, with

p1n = p0, for any n ∈ N∗ which converges to p0, with respect to the distance function dS

on S, and another sequence(p2n)n∈N∗ in B

2 (p0; ε0)\(B2 (p0; ε0) ∩W0

S), with p2n = p0,

for any n ∈ N∗ which converges to p0, with respect to the same dS . Since W0 and

B2 (p0; ε0) \(B2 (p0; ε0) ∩W0

S)are open in S, grad f is dierent from zero at any

point of W0 and grad f vanishes at any point of B2 (p0; ε0) \(B2 (p0; ε0) ∩W0

S), we

obtain a contradiction as in the proof of Theorem 3.16.


Second, let us consider that there exists ε0 > 0 such that grad f is dierent from zero

at any point ofB2 (p0; ε0)\(B2 (p0; ε0) ∩W0

S). ThenB2 (p0; ε0)\

(B2 (p0; ε0) ∩W0

S)⊂

S−C0

⊂ S, where S−C0

is the surface obtained by symmetry of S+C0

with respect to the

plane where its boundary lies. Clearly, SC0⊂ S. Since SC0

is complete, then it cannot

be extendible, and thus SC0= S.

In the last case, assume that for any εn > 0, in B2 (p0; εn) \(B2 (p0; εn) ∩W0

S)

there exists at least a point p1n such that (grad f)(p1n)= 0 and at least a point p2n such

that (grad f)(p2n)= 0.

Let us consider an arbitrary ε1 > 0. Then there exists U1 an open subset of S which

contains p21, which is connected, U1 ⊂ B2 (p0; ε1) \(B2 (p0; ε1) ∩W0

S)

and grad f

does not vanish at any point of U1. If we consider the connected component of W

which contains U1, one can notice that this is a surface SC10⊂ S and SC1

0

S= SC1

0

SC10 .

Moreover, SC10∩ S+

C0= ∅. We note that the boundaries of any two such connected

components coincide or they are disjoint.

Also, it is easy to see that the boundary of SC10does not intersect the boundary of

S+C0. Indeed, if these two boundaries would intersect, they would coincide and SC1

0=

S−C0. Therefore, we obtain a contradiction with our assumption.

Next, we can consider ε2 > 0 such that B2 (p0; ε2) \(B2 (p0; ε2) ∩W0

S)does not

intersect the boundary of SC10. With the same argument, one obtains another surface

SC20⊂ S such that SC2

0∩ S+

C0= ∅ and SC2

0∩ SC1

0= ∅. Moreover, SC2

0

SC20 ∩ S+

C0

SC0 = ∅

and SC20

SC20 ∩SC1

0

SC10 = ∅. We continue the reasoning and obtain a sequence of surfaces(

SCn0

)n∈N∗

⊂ S, which are disjoint two by two and SCn0∩ SC0

= ∅, for any n ∈ N∗.

Their boundaries are disjoint two by two and they do not intersect the boundary of

S+C0. For any n ∈ N∗, the boundary of SCn

0is a circle of radius

(Cn0

)−3/2= 2/

(3fCn

0

),

where fCn0is the mean curvature functions of SCn

0and is evaluated at some point of

the boundary of SCn0. But fCn

0= f on SCn

0

SCn0 , where f is the mean curvature function

of S. As f is continuous, the radius(Cn0

)−3/2converges to the radius of the circle

that gives the boundary of S+C0, which is

(C0

)−3/2= 2/

(3fC0

), where fC0

is the mean

curvature function of SC0and is evaluated at some point of the boundary of SC0

.

Therefore, we obtain a contradiction with the fact that S is a regular surface.


3.2 Complete biconservative surfaces in S3

As in the previous section, we consider the global problem for biconservative surfaces in

S3, i.e., our aim is to construct complete biconservative surfaces in S3, with grad f = 0

at any point of an open and dense subset.

We start with the following local extrinsic result.

Theorem 3.23 ([15]). Let M2 be a biconservative surface in S3 with (grad f)(p) = 0

at any point p ∈M . Then, locally, the surface, viewed in R4, can be parametrized by

YC1(u, v) = σ(u) +

4κ(u)−3/4

3√C1

(f1(cos v − 1) + f2 sin v

), (3.10)

where C1 ∈(64/

(35/4

),∞)is a positive constant; f1, f2 ∈ R4 are two constant or-

thonormal vectors; σ(u) is a curve parametrized by arc-length that satises

⟨σ(u), f1⟩ =4κ(u)−3/4

3√C1

, ⟨σ(u), f2⟩ = 0, (3.11)

and, as a curve in S2, its curvature κ = κ(u) is a positive non-constant solution of the

following ODE

κ′′κ =7

4

(κ′)2

+4

3κ2 − 4κ4 (3.12)

such that (κ′)2

= −16

9κ2 − 16κ4 + C1κ

7/2. (3.13)

Remark 3.24. The curve σ lies in the totally geodesic S2 = S3 ∩ Π, where Π is the

linear hyperspace of R4 orthogonal to f2.

Remark 3.25. The constant C1 determines uniquely the curvature κ, up to a transla-

tion of u, and then κ, f1 and f2 determine uniquely the curve σ.

In the following, we prove, following a slightly dierent method from that in [15],

that such a curve σ exists and nd a more explicit expression for (3.10).

Replacing (3.13) in (3.12), since κ′ = 0, we get

κ′′ = −16

9κ− 32κ3 +

7

4C1κ

5/2.

We consider f1 = e3 and f2 = e4, where e1, e2, e3, e4 is the canonical basis of R4.

From (3.11) it follows that σ can be written as

σ(u) =

(x(u), y(u),

4

3√C1

κ(u)−3/4, 0

).

3.2. Complete biconservative surfaces in S3 69

Using polar coordinates, we have x(u) = R(u) cosµ(u) and y(u) = R(u) sinµ(u), with

R(u) > 0.

Since σ(u) ⊂ S3, R2 = x2 + y2 and R > 0, we get κ >(16/

(9C1

))2/3and

R =

√1− 16

9C1

κ−3/2. (3.14)

As κ′(u) = 0, we can view u as a function of κ, and considering R = R(u(κ)) and

µ = µ(u(κ)), by a straightforward computation, it follows that σ is explicitly given by

σ(κ) =

(R cosµ,R sinµ,

4

3√C1

κ−3/4, 0

),

where R is given by (3.14) and

µ(κ) = ±

108

∫ κ

κ0

√C1τ

3/4(−16 + 9C1τ3/2

)√9C1τ3/2 − 16 (1 + 9τ2)

dτ + c1

,

where c1 is a real constant.

If we use the formula of σ in (3.10), we get

YC1(κ, v) =

(√1−

(4

3√C1

κ−3/4

)2

cosµ(κ),

√1−

(4

3√C1

κ−3/4

)2

sinµ(κ),

4

3√C1

κ−3/4 cos v, 4

3√C1

κ−3/4 sin v

).

(3.15)

Next, we have to determine the maximum domain for YC1. From (3.13), we ask

that −16κ2/9− 16κ4 + C1κ7/2 > 0. Since κ > 0, it is enough to nd the interval where

−16/9− 16κ2 + C1κ3/2 > 0. We denote by

L(κ) = −16

9− 16κ2 + C1κ

3/2, κ > 0.

We can see that if C1 > 64/(35/4), one obtains that there exist exactly two κ01 ∈(0,(3C1/64

)2)and κ02 ∈

((3C1/64

)2,∞)

such that L(κ01) = L(κ02) = 0 and

L(κ) > 0 for any κ ∈ (κ01, κ02).

We note that κ01 >(16/

(9C1

))2/3.

Therefore, the domain of YC1is (κ01, κ02)×R, where κ01 and κ02 are the vanishing

points of L, with 0 < κ01 < κ02.

We note that an alternative expression for YC1was given in [30].

Remark 3.26. We can choose c1 = 0 in the above expression of µ, by considering a

linear orthogonal transformation of R4.


We denote by

µ0(κ) = 108

∫ κ

κ0

√C1τ

3/4(−16 + 9C1τ3/2

)√9C1τ3/2 − 16 (1 + 9τ2)

dτ,

and, therefore, µ(κ) = ± (µ0(κ) + c1).

The following remark can be proved in a similar way as Lemma 3.35.

Remark 3.27. We have

limκκ01

µ0(κ) = µ0,−1 > −∞ and limκκ02

µ0(κ) = µ0,1 <∞.

Remark 3.28. For simplicity, we choose κ0 = (3C1/64)2.

If we denote S±C1,c1

the image of YC1, then we note that the boundary of S±

C1,c1is made up from two circles and along the boundary, the mean curvature function

is constant (two dierent constants) and its gradient vanishes. More precisely, the

boundary of S±C1,c1

is given by the curves(√1−

(4

3√C1

κ−3/401

)2

cosµ (κ01) ,

√1−

(4

3√C1

κ−3/401

)2

sinµ (κ01) ,

4

3√C1

κ−3/401 cos v, 4

3√C1

κ−3/401 sin v

)and (√

1−(

4

3√C1

κ−3/402

)2

cosµ (κ02) ,

√1−

(4

3√C1

κ−3/402

)2

sinµ (κ02) ,

4

3√C1

κ−3/402 cos v, 4

3√C1

κ−3/402 sin v

),

where µ (κ01) = limκκ01 µ(κ) and µ (κ02) = limκκ02 µ(κ) are real numbers.

These curves are circles in ane planes in R4 parallel to the Ox3x4 plane and their

radii are(4κ

−3/401

)/(3√C1

)and

(4κ

−3/402

)/(3√C1

), respectively.

At a boundary point, using the coordinates (µ, v), we get that the tangent plane to

the closure of S±C1,c1

in R4 is spanned by a vector which is tangent to the corresponding

circle and by−

√1−

(4

3√C1

κ−3/40i

)2

sinµ (κ0i) ,

√1−

(4

3√C1

κ−3/40i

)2

cosµ (κ0i) , 0, 0

,

where i = 1 or i = 2.

In a similar way to the proof of Theorem 3.4, we can obtain the next result.


Theorem 3.29. If φ : M2 → S3 is a biconservative surface with grad f = 0 at any

point, then there exists a unique C1 such that φ(M) ⊂ S±C1,c1

.

Thus, in order to construct, from extrinsic point of view, a complete biconservative

surface in S3, we can expect to glue along the boundary two biconservative surfaces of

type S±C1,c1,k

and S±C1,c1,l

, corresponding to the same C1, where k, l ∈ Z. In fact, if we

want to glue two surfaces corresponding to C1 and C ′1 along the boundary, then these

constants have to coincide and there is no ambiguity concerning along which circle of

the boundary we should glue the two pieces.

Geometrically, we start with a piece of type S+C1,0

corresponding to c1,0 = 0 and to

the sign +, and then consider T1(S+C1,0

), where T1 is a linear orthogonal transformation

of R4 that acts on R2 = span e1, e2 as an axial symmetry with respect to the line

determined by√√√√1−

(4

3√C1

κ−3/402

)2

cosµ0,1,

√√√√1−

(4

3√C1

κ−3/402

)2

sinµ0,1

and the origin, and leaves invariant span e3, e4. Moreover, T1 is a symmetry with

respect to the 3-dimensional subspace spanned by the vectors√√√√1−

(4

3√C1

κ−3/402

)2

cosµ0,1,

√√√√1−

(4

3√C1

κ−3/402

)2

sinµ0,1, 0, 0

,

e3 and e4. Of course, T1 leaves invariant the upper circle from the boundary of S+C1,0

.

The matrix of T1 is cos c1,1 sin c1,1 0 0

sin c1,1 − cos c1,1 0 0

0 0 1 0

0 0 0 1

.

We perform this process innitely many times. But it is dicult to conclude from

here that we get a complete biconservative surface in S3. From this process we obtain

a closed subset of S3 with self-intersections, but we cannot see if the surface that

we have obtained is the image of an isometric immersion (by composing with Tk, thedomain of YC1

does not change).

Of course, as a subset, the surface is complete with respect to the induced distance

from S3, but we want to show that the surface is complete with respect to the intrinsic

distance.

This construction of the complete biconservative surfaces in S3 can be illustrated in

R3 using the stereographic projection of S3, as in Figures 3.5 and 3.6.


Figure 3.5: Using the stereographic projection from the North pole.

Figure 3.6: Using the stereographic projection from (1, 0, 0, 0).

Further, as in the R3 case, we change the point of view and use the local intrinsic

characterization of the biconservative surfaces in S3.Another way to see that in the c = 0 case we have only a one-parameter family

of solutions of equation (2.30) is to rewrite the metric g in certain non-isothermal

coordinates. Further, we consider only the c = 1 case.


)be an abstract surface with g = e2ρ(u)(du2+dv2), where

u = u(ρ) satises

u =

∫ ρ

ρ0

dτ√−3be−2τ/3 − e2τ + a

+ u0,

where ρ is in some open interval I, a, b ∈ R are positive constants, and u0 ∈ R is a

constant. Then(M2, g

)is isometric to(

DC1 , gC1 =3

ξ2(−ξ8/3 + 3C1ξ2 − 3

)dξ2 + 1

ξ2dθ2

),

where DC1 = (ξ01, ξ02)×R, C1 ∈(4/(33/2

),∞)is a positive constant, and ξ01 and ξ02

are the positive vanishing points of −ξ8/3 + 3C1ξ2 − 3, with 0 < ξ01 < ξ02.


Proof. Since

u = u(ρ) =

∫ ρ

ρ0

dτ√−3be−2τ/3 − e2τ + a

+ u0,

we have that

du =1√

−3be−e−2ρ/3 − e2ρ + a

dρ,

and the metric g(u, v) = e2ρ(u)(du2 + dv2) can be rewritten as

g(ρ, v) =e2ρ

−3be−e−2ρ/3 − e2ρ + a

dρ2 + e2ρdv2.

if we consider the change of coordinates (ρ, v) =(log(33/4b3/8/ξ

), v), one obtains that

g(ξ, v) =1

ξ2

(3

−ξ8/3 + 3−1/2ab−3/4ξ2 − 3dξ2 + 33/2b3/4dv2

).

Now, considering another change of coordinates (ξ, v) =(ξ, 3−3/4b−3/8θ

)and denoting

C1 = 3−3/2ab−3/4 > 0, we obtain

g(ξ, θ) =1

ξ2

(3

−ξ8/3 + 3C1ξ2 − 3dξ2 + dθ2

),

for every ξ ∈ J , where J is an open interval such that −ξ8/3 + 3C1ξ2 − 3 > 0, for any

positive ξ ∈ J and C1 is a positive constant.

Next, we determine the interval J . If we denote

T (ξ) = −ξ8/3 + 3C1ξ2 − 3, ξ > 0,

we can see that its derivative is T ′(ξ) = −8ξ5/3/3 + 6C1ξ and it vanishes for ξ =

(9C1/4)3/2. The value of T at this critical point is 37C4

1/44 − 3. It is easy to see

that T is strictly increasing on(0, (9C1/4)

3/2), strictly decreasing on

((9C1/4)

3/2 ,∞)

and limξ→0 T (ξ) = −3, limξ→∞ T (ξ) = −∞. We ask T to have positive values, so we

have to determine when 37C41/4

4 − 3 > 0. It is easy to see that 37C41/4

4 − 3 > 0 if

C1 ∈(4/(33/2

),∞).

Therefore, T (ξ) > 0 for any ξ ∈ (ξ01, ξ02), where T (ξ01) = T (ξ02) = 0,

ξ01 ∈

(0,

(9

4C1

)3/2)

and ξ02 ∈

((9

4C1

)3/2

,∞

)(3.16)

are the only positive vanishing points of T and C1 ∈(4/(33/2

),∞).

Thus,(M2, g

)is isometric to(DC1 , gC1 =

3

ξ2(−ξ8/3 + 3C1ξ2 − 3

)dξ2 + 1

ξ2dθ2

),

where DC1 = (ξ01, ξ02) × R, C1 ∈(4/(33/2

),∞), and ξ01 and ξ02 are the vanishing

points of −ξ8/3 + 3C1ξ2 − 3, with 0 < ξ01 < ξ02.


The surface (DC1 , gC1) is not complete but it has the following properties.

Theorem 3.31. Consider (DC1 , gC1). Then, we have

(i) KC1(ξ, θ) = K(ξ, θ),

1−K(ξ, θ) =1

9ξ8/3 > 0, K ′(ξ) = − 8

27ξ5/3

and gradK = 0 at any point of DC1;

(ii) the immersion ϕC1 : (DC1 , gC1) → S3 given by

ϕC1(ξ, θ) =

(√1− 1

C1ξ2cos ζ(ξ),

√1− 1

C1ξ2sin ζ(ξ),

cos(√C1θ)√

C1ξ,sin(

√C1θ)√C1ξ

),

is biconservative in S3, where

ζ(ξ) = ±

∫ ξ

ξ00

√C1τ

4/3

(−1 + C1τ2)√−τ8/3 + 3C1τ2 − 3

dτ + c1

,

with c1 ∈ R a constant and ξ00 ∈ (ξ01, ξ02).

Proof. Consider the Riemannian metric

gC1 =3

ξ2(−ξ8/3 + 3C1ξ2 − 3)dξ2 +

1

ξ2dθ2

on DC1 with coecients given by

EC1 =3

ξ2(−ξ8/3 + 3C1ξ2 − 3), FC1 = 0, GC1 =

1

ξ2. (3.17)

Using the formula of the Gaussian curvature

K(ξ, θ) = − 1

2√EG

(∂

∂ξ

(Gξ√EG

)+∂

∂θ

(Eθ√EG

)),

we obtain that KC1 is given by

KC1(ξ, θ) = KC1(ξ) = −1

9ξ8/3 + 1

and

K ′C1(ξ) = − 8

27ξ5/3.

Therefore, K ′C1(ξ) < 0 at any ξ ∈ (ξ01, ξ02). Since

(gradKC1)(ξ, θ) =ξ2(−ξ8/3 + 3C1ξ

2 − 3)

3K ′C1(ξ)∂ξ,

we have that |(gradKC1) (ξ, θ)| = 0 for any (ξ, θ) ∈ DC1 .


To prove (ii), let us rst recall that, if M2 is a biconservative surface in S3, withgrad f = 0 at any point of M , then, as we have already seen in (3.15), M can be locally

parameterized by

YC1(κ, v) =

(√1−

(4

3√C1

κ−3/4

)2

cosµ(κ),

√1−

(4

3√C1

κ−3/4

)2

sinµ(κ),

4

3√C1

κ−3/4 cos v, 4

3√C1

κ−3/4 sin v

),

for any (κ, v) ∈ (κ01, κ02)×R, where κ01 and κ02 are the vanishing points of −16κ2/9−

16κ4 + C1κ7/2, κ01 ∈

(0,(3C1/64

)2), κ02 ∈

((3C1/64

)2,∞), C1 > 64/

(35/4

), and

µ(κ) = ±

108

∫ κ

κ0

√C1τ

3/4(−16 + 9C1τ3/2

)√9C1τ3/2 − 16 (1 + 9τ2)

dτ + c1

,

where c1 is a real constant.

In order to compute the metric on this surface, we need the coecients of the rst

fundamental form

EC1(κ, v) =

81C1κ3/2 − 144

κ2(9C1κ3/2 − 16

)(9C1κ3/2 − 16 (1 + 9κ2)

) ,FC1

(κ, v) = 0,

GC1(κ, v) =

16

9C1κ3/2.

Thus, the Riemannian metric is given by

gC1(κ, v) =

81C1κ3/2 − 144

κ2(9C1κ3/2 − 16

)(9C1κ3/2 − 16 (1 + 9κ2)

)dκ2 + 16

9C1κ3/2dv2.

We write C1 as C1 = 16 ·31/4C1, where C1 ∈ R∗+, and we know that C1 > 64/

(35/4

),

which implies C1 > 4/(33/2

). Therefore, we can choose C1 to be exactly the positive

constant from the metric (DC1 , gC1).

We note that we can consider the change of coordinates

(κ, v) =(3−3/2ξ4/3,

√C1θ

),

where ξ and θ are the coordinates on the domain DC1 . Indeed, we have

−ξ8/3 + 3C1ξ2 − 3 =

27

16κ2

(−16

9κ2 − 16κ4 + C1κ

7/2

)


and, therefore, the vanishing points ξ01 and ξ02 of−ξ8/3+3C1ξ2−3 are the corresponding

points to κ01 and κ02, i.e., ξ01 = 39/8κ3/401 and ξ02 = 39/8κ

3/402 .

Thus, we get the expression of the initial metric

gC1(ξ, θ) =3

ξ2(−ξ8/3 + 3C1ξ2 − 3

)dξ2 + 1

ξ2dθ2, (ξ, θ) ∈ DC1 .

Then, we dene ϕC1 as

ϕC1(ξ, θ) = Y31/4·16C1

(3−3/2ξ4/3,

√C1θ

).

Therefore,

ϕC1(ξ, θ) =

(√1− 1

C1ξ2cos ζ,

√1− 1

C1ξ2sin ζ,

cos(√C1θ)√

C1ξ,sin(

√C1θ)√C1ξ

),

for any ξ ∈ (ξ01, ξ02) and θ ∈ R, where ζ = µ(κ(ξ)) is given by

ζ(ξ) = ±

∫ ξ

ξ00

√C1τ

4/3

(−1 + C1τ2)√−τ8/3 + 3C1τ2 − 3

dτ + c1

,

where c1 is a real constant and we write ±c1 from esthetic reasons, as we will see

later.

Remark 3.32. For simplicity, we choose ξ00 = (9C1/4)3/2.

Remark 3.33. We note that the expression of the Gaussian curvature of (DC1 , gC1)

does not depend on C1. More precisely,

KC1(ξ, θ) = −1

9ξ8/3 + 1.

But, if we change further the coordinates (ξ, θ) =(ξ01 + ξ (ξ02 − ξ01) , θ

), then we x

the domain, i.e., (DC1 , gC1) is isometric to ((0, 1), gC1) and C1 appears in the expression

of KC1

(ξ, θ).

Remark 3.34. Since (gradKC1) (ξ, θ) = −(8ξ11/3

(−ξ8/3 + 3C1ξ

2 − 3)/81)∂ξ for any

(ξ, θ) ∈ DC1 , we get that

limξξ01

(gradKC1) (ξ, θ) = limξξ02

(gradKC1) (ξ, θ) = 0.

We denote by

ζ0(ξ) =

∫ ξ

ξ00

√C1τ

4/3

(−1 + C1τ2)√

−τ8/3 + 3C1τ2 − 3dτ

and we state the the following lemma that we will use later.


Lemma 3.35. We have

limξξ01

ζ0(ξ) = ζ0,−1 > −∞ and limξξ02

ζ0(ξ) = ζ0,1 <∞.

Proof. Let us dene the continuous functions

H(ξ) =√C1ξ

4/3 and G(ξ) =1

−1 + C1ξ2,

for any ξ ∈ [ξ01, ξ02] and C1 ∈(4/33/2,∞

). It easy to see that these functions have a

maximum and a minimum, which we denote by MH and mH respectively, for H, and

with MG and mG respectively, for G. Of course, mH , mG, MH , and MG are positive

constants.

Let us assume that ξ < ξ00. Then ζ0(ξ) < 0 for any ξ ∈ (ξ01, ξ00) and it is easy to

see that

ζ0(ξ) ≥MHMG

∫ ξ

ξ00

1√−τ8/3 + 3C1τ2 − 3

dτ.

Thus, to show that limξξ01 ζ0(ξ) is nite it suces to prove that

limξξ01

∫ ξ

ξ00

1√−τ8/3 + 3C1τ2 − 3

dτ > −∞.

To prove this, we consider the smooth function T (ξ) = −ξ8/3 + 3C1ξ2 − 3, for any

ξ ∈ (0,∞). We have

T (ξ) = T (ξ01) + T ′ (ξ01) (ξ − ξ01) + (ξ − ξ01)µ1(ξ), ξ > 0, (3.18)

where µ1 is a continuous function on (0,∞) such that limξ→ξ01 µ1(ξ) = 0.

We replace T (ξ01) = 0 and T ′ (ξ01) = −8ξ5/301 /3 + 6C1ξ01 in (3.18) and obtain

T (ξ) =

(−8

3ξ5/301 + 6C1ξ01

)(ξ − ξ01) + (ξ − ξ01)µ1(ξ)

= (ξ − ξ01)

(−8

3ξ5/301 + 6C1ξ01 + µ1(ξ)

), ξ > 0.

We note that(−8ξ

5/301 /3 + 6C1ξ01

)(ξ − ξ01) > 0 for ξ ∈ (ξ01, ξ00). Indeed, this is

equivalent to −8ξ5/301 /3 + 6C1ξ01 > 0, i.e., ξ01 < (9C1/4)

3/2, which, as we have already

seen, it is true.

Now, we consider β1(ξ) = 1/

√(−8ξ

5/301 /3 + 6C1ξ01

)(ξ − ξ01) and γ(ξ) = 1/

√T (ξ),

for any ξ ∈ (ξ01, ξ00). Since limξξ01 (γ(ξ)/β1(ξ)) = 1 ∈ (0,∞) and

limξξ01

∫ ξ

ξ00

β1(τ)dτ = limε→0

∫ ξ01−ε

ξ00

β1(τ) dτ = −2

√ξ00 − ξ01

−83ξ

5/301 + 6C1ξ01

∈ R∗−,


we get that limξξ01

∫ ξξ00γ(τ)dτ > −∞.

Therefore, limξξ01 ζ0(ξ) = ζ0,−1 > −∞.

Further, let us assume that ξ ≥ ξ00. Then ζ0(ξ) ≥ 0 for any ξ ∈ [ξ00, ξ02) and it is

also easy to see that

ζ0(ξ) ≤MHMG

∫ ξ

ξ00

1√−τ8/3 + 3C1τ2 − 3

dτ.

Thus, to prove that limξξ02 ζ0(ξ) is nite it suces to show that

limξξ02

∫ ξ

ξ00

1√−τ8/3 + 3C1τ2 − 3

dτ <∞.

In order to prove this, we again consider the smooth function T (ξ) = −ξ8/3+3Cξ2− 3,

for any ξ ∈ (0,∞) and we have

T (ξ) = T (ξ02) + T ′ (ξ02) (ξ − ξ02) + (ξ − ξ02)µ2(ξ), ξ > 0, (3.19)

where µ2 is a continuous function on (0,∞) such that limξ→ξ02 µ2(ξ) = 0.

We also have T (ξ02) = 0 and T ′ (ξ02) = −8ξ5/302 /3 + 6C1ξ02, and replacing in (3.19)

one obtains

T (ξ) =

(−8

3ξ5/302 + 6C1ξ02

)(ξ − ξ02) + (ξ − ξ02)µ2(ξ)

= (ξ − ξ02)

(−8

3ξ5/302 + 6C1ξ02 + µ2(ξ)

), ξ > 0.

We note that(−8ξ

5/302 /3 + 6C1ξ02

)(ξ − ξ02) > 0 for ξ ∈ [ξ00, ξ02). Indeed, this is

equivalent to −8ξ5/302 /3 + 6C1ξ02 < 0, i.e., ξ02 > (9C1/4)

3/2, which we have seen that is

true.

Now, we consider β2(ξ) = 1/

√(−8ξ

5/302 /3 + 6C1ξ02

)(ξ − ξ02) and again γ(ξ) =

1/√T (ξ), for any ξ ∈ [ξ00, ξ02). Since limξξ02 (γ(ξ)/β2(ξ)) = 1 ∈ (0,∞) and

limξξ02

∫ ξ

ξ00

β2(τ)dτ = limε→0

∫ ξ02−ε

ξ00

β2(τ)dτ = 2

√ξ00 − ξ02

−83ξ

5/302 + 6C1ξ02

∈ R∗+,


∫ ξξ00γ(τ)dτ <∞, which shows that limξξ02 ζ0(ξ) = ζ0,1 <∞.

Remark 3.36. The immersion ϕC1 depends on the sign ± and on the constant c1 in

the expression of ζ. As the classication is up to isometries of S3, the sign and the

constant are not important, but they play an important role in the gluing process.

The following result shows that we do have a one-parameter family of Riemannian

surfaces (DC1 , gC1).


Proposition 3.37. Let us consider(DC1 , gC1 =

3

ξ2(−ξ8/3 + 3C1ξ2 − 3

)dξ2 + 1

ξ2dθ2

)and DC′

1, gC′

1=

3

ξ2(−ξ8/3 + 3C ′

1ξ2 − 3

)dξ2 + 1

ξ2dθ2

.

The surfaces (DC1 , gC1) and(DC′

1, gC′

1

)are isometric if and only if C1 = C ′

1 and the

isometry is Θ(ξ, θ) = (ξ,±θ + constant). Therefore, we have a one-parameter family of

surfaces.

Proof. Assume that there exists an isometry Θ : (DC1 , gC1) →(DC′

1, gC′

1

)and denote

Θ(ξ, θ) =(Θ1(ξ, θ),Θ2(ξ, θ)

). As we have seen in Theorem 3.31, the Gaussian curvature

of (DC1 , gC1) is K(ξ, θ) = −ξ8/3/9 + 1 and the Gaussian curvature of(DC′

1, gC′

1

)is

K(ξ, θ) = −ξ8/3/9 + 1.

Since Θ is an isometry, we have that K(Θ(ξ, θ)) = K(ξ, θ) and, taking into account

the above expressions of the curvatures, we get Θ1(ξ, θ) = ξ > 0. Therefore, Θ(ξ, θ) =(ξ,Θ2(ξ, θ)

).

Next, from(Θ∗gC′

1

)(∂ξ, ∂ξ) = gC1 (∂ξ, ∂ξ), i.e., gC′

1(Θ∗∂ξ,Θ∗∂ξ) = gC1 (∂ξ, ∂ξ),

using (3.17), we nd

3

−ξ8/3 + 3C1ξ2 − 3=

3

−ξ8/3 + 3C ′1ξ

2 − 3+

(∂Θ2

∂ξ

)2

. (3.20)

Similarly, from(Θ∗gC′

1

)(∂ξ, ∂θ) = gC1 (∂ξ, ∂θ) and

(Θ∗gC′

1

)(∂θ, ∂θ) = gC1 (∂θ, ∂θ), us-

ing (3.17), we get

0 =∂Θ2

∂ξ· ∂Θ

2

∂θand

∂Θ2

∂θ= ±1. (3.21)

From (3.21) one obtains ∂Θ2

∂ξ = 0. Now, using (3.20), it follows that C1 = C ′1. Since

∂Θ2

∂ξ = 0 and ∂Θ2

∂θ = ±1, we have Θ(ξ, θ) = (ξ,±θ+ a1), where a1 is a real constant.

The construction, from intrinsic point of view, of complete biconservative surfaces

in S3 consists in two steps, and the key idea is to notice that (DC1 , gC1) is, locally and

intrinsically, isometric to a surface of revolution in R3.

The rst step is to construct a complete surface of revolution in R3 which on an

open dense subset is locally isometric to (DC1 , gC1). We start with the next result.

Theorem 3.38. Let us consider (DC1 , gC1) as above. Then (DC1 , gC1) is the universal

cover of the surface of revolution in R3 given by

ψC1,C∗1(ξ, θ) =

(χ(ξ) cos

θ

C∗1

, χ(ξ) sinθ

C∗1

, ν(ξ)

), (3.22)


where χ(ξ) = C∗1/ξ,

ν(ξ) = ±

∫ ξ

ξ00

√3τ2 − (C∗

1 )2 (−τ8/3 + 3C1τ2 − 3

)τ4(−τ8/3 + 3C1τ2 − 3

) dτ + c∗1, (3.23)

C∗1 ∈

(0,√(

33/2)/(33/2C1 − 4

) )is a positive constant and c∗1 ∈ R is constant.

Proof. In fact, we can prove that if (DC1 , gC1) is (locally and intrinsically) isometric to

a surface of revolution, then it has to be of form (3.22). To show this, let us consider

ψ(ξ, θ)=(χ(ξ)cos θ, χ

(ξ)sin θ, ν

(ξ))

,(ξ, θ)∈ D,

a surface of revolution, where D is an open set in R2 and Θ : (DC1 , gC1) →(D, g

)an

isometry, where

g(ξ, θ)=

((χ′(ξ))2

+(ν ′(ξ))2)

dξ2 +(χ(ξ))2

dθ2.

We assume that χ(ξ)> 0 for any ξ.

Next, we proceed in the same way as in the proof of Proposition 3.37. From

K(Θ(ξ, θ)) = K(ξ, θ), we get Θ1(ξ, θ) = Θ1(ξ). In order to simplify the notations,

we write Θ1 = ξ and Θ2 = θ, so that ξ(ξ, θ) = ξ(ξ). As Θ∗g = gC1 , we get(∂θ

∂θ

)2 (χ(ξ(ξ)

))2=

1

ξ2(3.24)

and∂θ

∂θ

∂θ

∂ξ

(χ(ξ(ξ)

))2= 0. (3.25)

From (3.24), one has ∂θ∂θ = 0, and then, from (3.25), it follows that ∂θ

∂ξ = 0. Thus we

have θ(ξ, θ) = θ(θ). Again from (3.24), one obtains(∂θ∂θ

)2= 1/

(ξ2(χ(ξ(ξ)

))2).

Since the left hand term depends only on θ and the right hand term depends only on

ξ, it follows that

χ(ξ(ξ)

)=C∗

ξ, (3.26)

where C∗ ∈ R∗+, and

θ(θ) =θ

C∗ + a0,

where a0 ∈ R. In the following, we shall consider a0 = 0.

Hence, we obtain((χ ξ

)′(ξ)

)2

+

((ν ξ

)′(ξ)

)2

=3

ξ2(−ξ8/3 + 3C1ξ2 − 3

)


and, from (3.26), one has((ν ξ

)′(ξ)

)2

=3ξ2 − (C∗

1 )2 (−ξ8/3 + 3C1ξ

2 − 3)

ξ2(−ξ8/3 + 3C1ξ2 − 3

) . (3.27)

Next, we have to nd the conditions to be satised by the positive constant C∗1 , such

that 3ξ2 − (C∗1 )

2 (−ξ8/3 + 3C1ξ2 − 3

)> 0 for any ξ ∈ (ξ01, ξ02), where C1 > 4/

(33/2

)is xed.

Let us denote

P (ξ) = 3ξ2 − (C∗1 )

2(−ξ8/3 + 3C1ξ

2 − 3), ξ ∈ [ξ01, ξ02] ,

We ask that P (ξ) > 0, for any ξ ∈ (ξ01, ξ02). The rst derivative of P is

P ′(ξ) = ξ

(8

3(C∗

1 )2 ξ2/3 − 6C1 (C

∗1 )

2 − 6

)and, we note that P ′(ξ) = 0 if and only if ξ2/3 = 9

(C1 (C

∗1 )

2 − 1)/(4 (C∗

1 )2).

If we choose C∗1 such that C1 (C

∗1 )

2 − 1 ≤ 0, i.e., C∗1 ∈

(0, 1/

√C1

), then P ′(ξ) = 0,

for any ξ ∈ (ξ01, ξ02). Since P (ξ01) = 3ξ201 > 0 and P (ξ02) = 3ξ202 > 0, we get that

P (ξ) > 0, for any ξ ∈ (ξ01, ξ02).

Now we study the case when C∗1 > 1/

√C1. We recall that in the proof of Proposition

3.30 we have seen that T (ξ) = −ξ8/3+3C1ξ2−3 is strictly increasing on

(0, (9C1/4)

3/2)

and T (ξ01) = 0, where ξ01 ∈(0, (9C1/4)

3/2). Here, we can take (0,∞) as the domain

of denition of T .

We note that the critical point(9(C1 (C

∗1 )

2 − 1)/(4 (C∗

1 )2))3/2

for P is in the

interval(0, (9C1/4)

3/2), for any C1 > 4/

(33/2

), and

T

9

(C1 (C

∗1 )

2 − 1)

4 (C∗1 )

2

3/2 = −

9(C1 (C

∗1 )

2 − 1)

4 (C∗1 )

2

4

+ 3C1

9(C1 (C

∗1 )

2 − 1)

4 (C∗1 )

2

3

− 3.

To simplify the expression, we denote (C∗1 )

2 by A and we have A > 1/C1. Now, we

dene a new function

S(A) = −(9(C1A− 1)

4A

)4

+ 3C1

(9(C1A− 1)

4A

)3

− 3, A >1

C1.

Using standard computations, we get that S is a strictly increasing function,

limA→1/C1

S(A) = −3 and limA→∞

S(A) =3

4

(9

4

)3

C41 + 3 > 0.

We denote by A ∈ (1/C1,∞) the point where S vanishes. Thus, S(A) < 0, for any

A ∈(1/C1, A

)and S(A) ≥ 0, for any A ∈ [A,∞). We split our study in two cases as

A ∈(1/C1, A

)and A ∈

[A,∞

).


If A ∈(1/C1, A

), since S is a strictly increasing function on (1/C1,∞), we have

T

((9(C1A− 1)

4A

)3/2)

= S(A) < S(A) = 0 = T (ξ01),

and therefore, since T is strictly increasing on the interval(0, (9C1/4)

3/2), and(

9(C1A− 1)

4A

)3/2

∈

(0,

(9C1

4

)3/2),

we get (9 (C1A− 1) /(4A))3/2 < ξ01. Thus, the point at which the rst derivative of P

vanishes is outside of the domain (ξ01, ξ02). As above, P is a strictly increasing function,

with positive values, for every ξ ∈ (ξ01, ξ02).

If A ≥ A, since S is a strictly increasing function on (1/C1,∞), we have

T

((9(C1A− 1)

4A

)3/2)

= S(A) ≥ S(A) = 0 = T (ξ01),

and therefore, since T is strictly increasing on the interval(0, (9C1/4)

3/2], and(

9(C1A− 1)

(4A)

)3/2

∈

(0,

(9C1

4

)3/2),

we get (9(C1A− 1)/(4A))3/2 ≥ ξ01.

We also have ξ02 > (9C1/4)3/2 > (9(C1A− 1)/(4A))3/2 and thus, we get that

(9(C1A− 1)/(4A))3/2 ∈ (ξ01, ξ02).

We want P to have positive values for any ξ ∈ (ξ01, ξ02), and, since the val-

ues of P at ξ01 and at ξ02 are 3ξ201 > 0 and 3ξ202 > 0, respectively, we have to

ask that P((9(C1A− 1)/(4A))3/2

)> 0. It is easy to see that this is equivalent to

A < 33/2/(33/2C1 − 4

).

Therefore, for A ∈[A, 33/2/

(33/2C1 − 4

)), we get that P (ξ) > 0, for any ξ ∈

(ξ01, ξ02).

Consequently, since A = (C∗1 )

2, for C∗1 ∈

(0,√

33/2/(33/2C1 − 4

)), we have P (ξ) >

0 for any ξ ∈ (ξ01, ξ02) and(ν ξ

)(ξ) = ±

∫ ξ

ξ00

√3τ2 − (C∗

1 )2 (−τ8/3 + 3C1τ2 − 3

)τ4(−τ8/3 + 3C1τ2 − 3

) dτ + c∗1,

for any ξ ∈ (ξ01, ξ02), where c∗1 is a real constant.

Next, we consider ψC1,C∗1= ψ Θ dened by

ψC1,C∗1(ξ, θ) =

((χ ξ

)(ξ) cos

(θ(θ)

),(χ ξ

)(ξ) sin

(θ(θ)

),(ν ξ

)(ξ))

=

(χ(ξ) cos

θ

C∗1

, χ(ξ) sinθ

C∗1

, ν(ξ)

), (ξ, θ) ∈ DC1 ,


where C1 > 4/(33/2

)is a positive constant, C∗

1 ∈(0,√(

33/2)/(33/2C1 − 4

) ), χ(ξ) =

C∗1/ξ and

ν(ξ) = ±∫ ξ

ξ00

√3τ2 − (C∗

1 )2 (−τ8/3 + 3C1τ2 − 3

)τ4(−τ8/3 + 3C1τ2 − 3

) dτ + c∗1,

for any ξ ∈ (ξ01, ξ02), with c∗1 a real constant.

Remark 3.39. The mean curvature function of ψC1,C∗1is given by

fC1,C∗1=

9ξ2 − (C∗1 )

2 (−2ξ8/3 + 9C1ξ2 − 18

)6C∗

1

√9ξ2 − 3 (C∗

1 )2 (−ξ8/3 + 3C1ξ2 − 3

)and we can see that it depends on both C1 and C

∗1 .

Remark 3.40. From now on, we will take ξ00 = (9C1/4)3/2 ∈ (ξ01, ξ02) and C∗

1 ∈(0,√(

33/2)/(33/2C1 − 4

) ).

The function ν has the following properties which follows easily.

Lemma 3.41. Let

ν0(ξ) =

∫ ξ

ξ00

√3τ2 − (C∗

1 )2 (−τ8/3 + 3C1τ2 − 3

)τ4(−τ8/3 + 3C1τ2 − 3

) dτ, ξ ∈ (ξ01, ξ02) ,

i.e., we x the sign in (3.23) and we choose c∗1 = c∗1,0 = 0. Then

(i) limξξ01 ν0(ξ) = ν0,−1 > −∞ and limξξ02 ν0(ξ) = ν0,1 <∞;

(ii) ν0 is strictly increasing and

limξξ01

ν ′0(ξ) = limξξ02

ν ′0(ξ) = ∞;

(iii) limξξ01 ν′′0 (ξ) = −∞ and limξξ02 ν

′′0 (ξ) = ∞.

Proof. We prove (i) in a similar way to the proof of Lemma 3.35. More precisely, we

consider the functions

H(ξ) =√3ξ2 − (C∗

1 )2 (−ξ8/3 + 3C1ξ2 − 3

)and G(ξ) =

1

ξ2,

with ξ ∈ [ξ01, ξ02]. It easy to see that these functions have a maximum and a min-

imum, which we denote by MH and mH , respectively, for H, and with MG and mG,

respectively, for G. Of course, mH , mG, MH , and MG are positive constants.

Let us assume that ξ < ξ00. Then ν0(ξ) < 0 for any ξ ∈ (ξ01, ξ00) and it is easy to

see that

ν0(ξ) ≥MHMG

∫ ξ

ξ00

1√−τ8/3 + 3C1τ2 − 3

dτ.


Thus, to show that limξξ01 ν0(ξ) is nite it is enough to prove that

limξξ01

∫ ξ

ξ00

1√−τ8/3 + 3C1τ2 − 3

dτ > −∞.

In order to prove this, we consider again the smooth function T (ξ) = −ξ8/3+3C1ξ2−3,

for any ξ ∈ (0,∞). We have

T (ξ) = T (ξ01) + T ′ (ξ01) (ξ − ξ01) + (ξ − ξ01)µ1(ξ), ξ > 0,

where µ1 is a continuous function on (0,∞) and limξ→ξ01 µ1(ξ) = 0.

Since T (ξ01) = 0 and T ′ (ξ01) = −8ξ5/301 /3 + 6C1ξ01 we obtain

T (ξ) =

(−8

3ξ5/301 + 6C1ξ01

)(ξ − ξ01) + (ξ − ξ01)µ1(ξ)

= (ξ − ξ01)

(−8

3ξ5/301 + 6C1ξ01 + µ1(ξ)

), ξ > 0.

We note that(−8ξ

5/301 /3 + 6C1ξ01

)(ξ − ξ01) > 0 for ξ ∈ (ξ01, ξ00). This is, indeed,

equivalent to −8ξ5/301 /3 + 6C1ξ01 < 0, i.e., ξ01 > (9C1/4)

3/2, which is true.

Now, we consider the functions β(ξ) = 1/

(√(−8ξ

5/301 /3 + 6C1ξ01

)(ξ − ξ01)

)and

γ(ξ) = 1/√

−ξ8/3 + 3C1ξ2 − 3, for any ξ ∈ (ξ01, ξ00). Since limξξ01 (γ(ξ)/β(ξ)) = 1 ∈(0,∞) and

limξξ01

∫ ξ

ξ00

β(τ)dτ = limε→0

∫ ξ01−ε

ξ00

β(τ)dτ = −2

√ξ00 − ξ01

−83ξ

5/301 + 6C1ξ01

∈ R∗−,


∫ ξξ00γ(τ) dτ > −∞, which means that limξξ01 ν0(ξ) = ν0,−1 >

−∞.

In the same way, we can prove that limξξ02

∫ ξξ00

1/√−τ8/3 + 3C1τ2 − 3 dτ < ∞,

and then

limξξ02

ν0(ξ) = ν0,1 <∞.

In order to prove (ii), we rst note that

ν ′0(ξ) =

√3ξ2 − (C∗

1 )2 (−ξ8/3 + 3C1ξ2 − 3

)ξ4(−ξ8/3 + 3C1ξ2 − 3

) > 0, ξ ∈ (ξ01, ξ02) ,

and, therefore, ν0 is a strictly increasing function and

limξξ01

ν ′0(ξ) = limξξ02

ν ′0(ξ) = ∞.


To prove (iii), we can rewrite the derivative of ν0 as

ν ′0(ξ) =

√3

ξ2(−ξ8/3 + 3C1ξ2 − 3

) − (C∗1 )

2

ξ4,

and by a straightforward computation we obtain that

ν ′′0 (ξ) =1

2√3ξ2 − (C∗

1 )2 (−ξ8/3 + 3C1ξ2 − 3

)(

1(−ξ8/3 + 3C1ξ2 − 3

)3/2 ·(−6(−ξ8/3 + 3C1ξ

2 − 3)

ξ− 3

(−8

3ξ5/3 + 6C1ξ

))+

+4 (C∗

1 )2√

−ξ8/3 + 3C1ξ2 − 3

ξ3

).

We have −8ξ5/301 /3+ 6C1ξ01 > 0 and −8ξ

5/302 /3+ 6C1ξ02 < 0 since these inequalities are

equivalent to relations (3.16). Therefore,

limξξ01

ν ′′0 (ξ) = −∞ and limξξ02

ν ′′0 (ξ) = ∞.

Remark 3.42. The immersion ψC1,C∗1depends on the sign ± and on the constant c∗1

in the expression of ν. We denote by S±C1,C∗

1 ,c∗1the image of ψC1,C∗

1.

We note that the boundary of S±C1,C∗

1 ,c∗1is given by the curves(

C∗1

ξ01cos

θ

C∗1

,C∗1

ξ01sin

θ

C∗1

, ν (ξ01)

)and (

C∗1

ξ02cos

θ

C∗1

,C∗1

ξ02sin

θ

C∗1

, ν (ξ02)

)These curves are circles in ane planes in R3 parallel to the Oxy plane and their radii

are C∗1/ξ01 and C

∗1/ξ02, respectively.

At a boundary point, using the coordinates (ν, θ), we get that the tangent plane to

the closure of S±C1,C∗

1 ,c∗1is spanned by a vector which is tangent to the corresponding

circle and by the vector (0, 0, 1). Thus, the tangent plane is parallel to the rotational

axis Oz.

Geometrically, we start with a piece of type S±C1,C∗

1 ,c∗1and by symmetry to the planes

where the boundary lie, we get our complete surface SC1,C∗1; the process is periodic and

we perform it along the whole Oz axis.


Analytically, we x C1 and C∗1 , and alternating the sign and with appropriate choices

of the constant c∗1, we can construct a complete surface of revolution SC1,C∗1in R3 which

on an open subset is locally isometric to (DC1 , gC1). In fact, these choices of + and −,and of the constants c∗1 are uniquely determined by the rst choice of +, or of −, andof the constant c∗1. We start with + and c∗1 = c∗1,0.

Further, we will give more details about this construction. Let us consider the

prole curve σ0(ξ) = (χ(ξ), ν0(ξ)), for any ξ ∈ (ξ01, ξ02). Obviously, ν0 : (ξ01, ξ02) →(ν0,−1, ν0,1) is a dieomorphism and we can consider ν−1

0 : (ν0,−1, ν0,1) → (ξ01, ξ02), with

ν−10 : ξ0 = ξ0(ν), ν ∈ (ν0,−1, ν0,1). One can reparametrize σ0 such that it becomes the

graph of a function depending on the variable ν, ν ∈ (ν0,−1, ν0,1).

In order to extend our surface to the upper part, we ask the line ν = ν0,1 to be a

symmetry axis. Therefore 2ν0,1 = ν0(ξ) + ν1(ξ), where ν1 : (ξ01, ξ02) → R, and then we

get ν1(ξ) = 2ν0,1 − ν0(ξ); thus c∗1 = c∗1,1 = 2ν0,1. It is easy to see that

limξξ01

ν1(ξ) = 2ν0,1 − ν0,−1, limξξ02

ν1(ξ) = ν0,1,

and, since ν ′1(ξ) = −ν ′0(ξ) < 0, for any ξ ∈ (ξ01, ξ02), it follows that ν1 is strictly

decreasing and ν1 (ξ01, ξ02) = (ν0,1, 2ν0,1 − ν0,−1). Since ν1 is a dieomorphism on its

image, we can consider ν−11 : (ν0,1, 2ν0,1 − ν0,−1) → (ξ01, ξ02), with ν−1

1 : ξ1 = ξ1(ν),

ν ∈ (ν0,1, 2ν0,1 − ν0,−1).

It is easy to see that

limνν0,1

ξ1(ν) = ξ02, limν2ν0,1−ν0,−1

ξ1(ν) = ξ01,

and, since(ν−11

)′(ν) = 1/ (ν ′1(ξ1(ν))) < 0, for any ν ∈ (ν0,1, 2ν0,1 − ν0,−1), it follows

that ν−11 is strictly decreasing.

Next, we dene a function F1 : (ν0,−1, 2ν0,1 − ν0,−1) → R by

F1(ν) =

ξ1(ν), ν ∈ (ν0,1, 2ν0,1 − ν0,−1)

ξ02, ν = ν0,1

ξ0(ν), ν ∈ (ν0,−1, ν0,1)

,

and we will prove that F1 is at least of class C3.

Obviously, F1 is continuous.

In order to prove that F1 is of class C1, rst we consider ν ∈ (ν0,−1, ν0,1). In this

case, we have

F ′1(ν) = ξ′0(ν) =

1

ν ′0(ξ0(ν))

and

limνν0,1

F ′1(ν) = lim

νν0,1ξ′0(ν) = lim

νν0,1

1

ν ′0(ξ0(ν))= lim

ξξ02

1

ν ′0(ξ)= 0.


Then, if we consider ν ∈ (ν0,1, 2ν0,1 − ν0,−1), one gets

F ′1(ν) = ξ′1(ν) =

1

ν ′1(ξ1(ν))

and

limνν0,1

F ′1(ν) = lim

νν0,1ξ′1(ν) = lim

νν0,1

1

ν ′1(ξ1(ν))= lim

ξξ02

1

ν ′1(ξ)= lim

ξξ02

1

−ν ′0(ξ)= 0.

Therefore, limνν0,1 F′1(ν) = limνν0,1 F

′1(ν) = 0 ∈ R, which means that there exists

F ′1(ν0,1) = 0 and F1 is of class C

1.

To prove that F1 is of class C2 we consider the same two situations. First assume

that ν ∈ (ν0,−1, ν0,1). In this case, one has

F ′′1 (ν) = −ν

′′0 (ξ0(ν)) · ξ′0(ν)(ν ′0(ξ0(ν)))

2= − ν ′′0 (ξ0(ν))

(ν ′0(ξ0(ν)))3

and

limνν0,1

F ′′1 (ν) = − lim

ξξ02

ν ′′0 (ξ)

(ν ′0(ξ))3.

From the denition of ν0, by a straightforward computation, we obtain the expressions

for the rst and second derivatives of ν0, and replacing them in the above relation we

get

limνν0,1

F ′′1 (ν) =

1

6ξ302

(−8

3ξ2/302 + 6C1

)∈ R.

Now, assume that ν ∈ (ν0,1, 2ν0,1 − ν0,−1). In this case

F ′′1 (ν) = ξ′′1 (ν) = −ν

′′1 (ξ1(ν)) · ξ′1(ν)(ν ′1(ξ1(ν)))

2= − ν ′′1 (ξ1(ν))

(ν ′1(ξ1(ν)))3

and

limνν0,1

F ′′1 (ν) = − lim

ξξ02

ν ′′1 (ξ)

(ν ′1(ξ))3= − lim

ξξ02

ν ′′0 (ξ)

(ν ′0(ξ))3.

Therefore,

limνν0,1

F ′′1 (ν) = lim

νν0,1F ′′1 (ν) =

1

6ξ302

(−8

3ξ2/302 + 6C1

)∈ R,

which means that there exists F ′′1 (ν0,1) =

16ξ

302

(−8

3ξ2/302 + 6C1

), and F1 is of class C

2.

Now, we prove that F1 is of class C3. First, assume ν ∈ (ν0,−1, ν0,1). In this case

F ′′′1 (ν) =

−ν ′′′0 (ξ0(ν)) · ν ′0(ξ0(ν)) + 3(ν ′′0 (ξ0(ν)))2

(ν ′0(ξ0(ν)))5

and

limνν0,1

F ′′′1 (ν) = lim

ξξ02

−ν ′′′0 (ξ) · ν ′0(ξ) + 3(ν ′′0 (ξ))2

(ν ′0(ξ))5

.


From the denition of ν0, the rst and second derivatives of ν0, we can compute the

third derivative of ν0, and substituting them in the above relation, one obtains

limνν0,1

F ′′′1 (ν) = 0.

The second case is when ν ∈ (ν0,1, 2ν0,1 − ν0,−1). In this case, we have

F ′′′1 (ν) =

−ν ′′′1 (ξ1(ν)) · ν ′1(ξ1(ν)) + 3(ν ′′1 (ξ1(ν)))2

(ν ′1(ξ1(ν)))5

and

limνν0,1

F ′′′1 (ν) = lim

ξξ02

−ν ′′′1 (ξ) · ν ′1(ξ) + 3(ν ′′1 (ξ))2

(ν ′1(ξ))5

= limξξ02

−(−ν ′′′0 (ξ)) · (−ν ′0(ξ)) + 3(−ν ′′0 (ξ))2

−(ν ′0(ξ))5

= − limξξ02

−ν ′′′0 (ξ) · ν ′0(ξ) + 3(ν ′′0 (ξ))2

(ν ′0(ξ))5

.

Therefore,

limνν0,1

F ′′′1 (ν) = lim

νν0,1F ′′′1 (ν) = 0 ∈ R,

which shows that there exists F ′′′1 (ν0,1) = 0 and F1 is of class C

3.

In order to extend our surface to the lower part, we ask the line ν = ν0,−1 to be a

symmetry axis. Therefore, 2ν0,−1 = ν0(ξ) + ν−1(ξ), where ν−1 : (ξ01, ξ02) → R, and we

get ν−1(ξ) = 2ν0,−1 − ν0(ξ); thus c∗1 = c∗1,−1 = 2ν0,−1. It is easy to see that

limξξ02

ν−1(ξ) = 2ν0,−1 − ν0,1, limξξ01

ν−1(ξ) = ν0,−1,

and, since ν ′−1(ξ) = −ν ′0(ξ) < 0, for any ξ ∈ (ξ01, ξ02), it follows that ν−1 is strictly

decreasing and ν−1 (ξ01, ξ02) = (2ν0,−1 − ν0,1, ν0,−1). Since ν−1 is a dieomorphism

on its image, one can consider ν−1−1 : (2ν0,−1 − ν0,1, ν0,−1) → (ξ01, ξ02), with ν−1

−1 :

ξ−1 = ξ−1(ν), ν ∈ (2ν0,−1 − ν0,1, ν0,−1).

It is easy to see that

limν2ν0,−1−ν0,1

ξ−1(ν) = ξ02, limνν0,−1

ξ−1(ν) = ξ01,

and, since(ν−1−1

)′(ν) = 1/

(ν ′−1(ξ−1(ν))

)< 0, for any ν ∈ (2ν0,−1 − ν0,1, ν0,−1), we get

that ν−1−1 is strictly decreasing.

Further, we dene the function F−1 : (2ν0,−1 − ν0,1, ν0,1) → R by

F−1(ν) =

ξ0(ν), ν ∈ (ν0,−1, ν0,1)

ξ01, ν = ν0,−1

ξ−1(ν), ν ∈ (2ν0,−1 − ν0,1, ν0,−1)

.


In a similar way to F1 case, we prove that F−1 is at least of class C3.

Obviously, F−1 is continuous.

To prove that F−1 is of class C1, we rst consider ν ∈ (ν0,−1, ν0,1). In this case

F ′−1(ν) = ξ′0(ν) =

1

ν ′0(ξ0(ν))

and

limνν0,−1

F ′−1(ν) = lim

νν0,−1

ξ′0(ν) = limνν0,−1

1

ν ′0(ξ0(ν))

= limξξ01

1

ν ′0(ξ)= 0.

Then, we consider ν ∈ (2ν0,−1 − ν0,1, ν0,−1). In this case

F ′−1(ν) = ξ′−1(ν) =

1

ν ′−1(ξ−1(ν))

and

limνν0,−1

F ′−1(ν) = lim

νν0,−1

ξ′−1(ν) = limνν0,−1

1

ν ′−1(ξ−1(ν))

= limξξ01

1

ν ′−1(ξ)= lim

ξξ01

1

−ν ′0(ξ)= 0.

Therefore, limνν0,−1 F′−1(ν) = limνν0,−1 F

′−1(ν) = 0 ∈ R, which means that there

exists F ′−1(ν0,−1) = 0 and F−1 is of class C

1.

In order to prove that F−1 is of class C2, we consider the same two situations. First,

let ν ∈ (ν0,−1, ν0,1). In this case, one has

F ′′−1(ν) = −ν

′′0 (ξ0(ν)) · ξ′0(ν)(ν ′0(ξ0(ν)))

2= − ν ′′0 (ξ0(ν))

(ν ′0(ξ0(ν)))3

and

limνν0,−1

F ′′−1(ν) = − lim

ξξ01

ν ′′0 (ξ)

(ν ′0(ξ))3.

By a straightforward computation, we can see that

limνν0,−1

F ′′−1(ν) = −1

6ξ301

(−8

3ξ2/301 + 6C1

)∈ R.

Second, assume that ν ∈ (2ν0,−1 − ν0,1, ν0,−1). Then

F ′′−1(ν) = ξ′−1(ν) = −

ν ′′−1(ξ−1(ν)) · ξ′−1(ν)

(ν ′−1(ξ−1(ν)))2= −

ν ′′−1(ξ−1(ν))

(ν ′−1(ξ−1(ν)))3

and

limνν0,−1

F ′′−1(ν) = − lim

ξξ01

ν ′′−1(ξ)

(ν ′−1(ξ))3= − lim

ξξ01

ν ′′0 (ξ)

(ν ′0(ξ))3.


Therefore,

limνν0,−1

F ′′−1(ν) = lim

νν0,−1

F ′′−1(ν) = −1

6ξ301

(−8

3ξ2/301 + 6C1

)∈ R,

which means that there exists F ′′−1(ν0,−1) = −(1/6)ξ301

(−8ξ

2/301 /3 + 6C1

)and F−1 is of

class C2.

Further, we prove that F−1 is of class C3. First, consider ν ∈ (ν0,−1, ν0,1). In this

case

F ′′′−1(ν) =

−ν ′′′0 (ξ0(ν)) · ν ′0(ξ0(ν)) + 3(ν ′′0 (ξ0(ν)))2

(ν ′0(ξ0(ν)))5

and

limνν0,−1

F ′′′−1(ν) = lim

ξξ01

−ν ′′′0 (ξ) · ν ′0(ξ) + 3(ν ′′0 (ξ))2

(ν ′0(ξ))5

= 0.

If ν ∈ (2ν0,−1 − ν0,1, ν0,−1), we have

F ′′′−1(ν) =

−ν ′′′−1(ξ−1(ν)) · ν ′−1(ξ−1(ν)) + 3(ν ′′−1(ξ−1(ν)))2

(ν ′−1(ξ−1(ν)))5

and

limνν0,−1

F ′′′−1(ν) = lim

ξξ01

−ν ′′′−1(ξ) · ν ′−1(ξ) + 3(ν ′′−1(ξ))2

(ν ′−1(ξ))5

= − limξξ01

−ν ′′′0 (ξ) · ν ′0(ξ) + 3(ν ′′0 (ξ))2

(ν ′0(ξ))5

.

Therefore,

limνν0,−1

F ′′′−1(ν) = lim

νν0,−1

F ′′′−1(ν) = 0 ∈ R,

which means that there exists F ′′′−1(ν0,−1) = 0 and F−1 is of class C

3.

Now, we extend the functions F1 and F−1 to the whole line R. This construction

will be done by symmetry to the lines ν = ν0,k, k ∈ Z∗.

We dene ν0,2 = 2ν0,1 − ν0,−1, ν0,3 = 2ν0,2 − ν0,1 = 3ν0,1 − 2ν0,−1, etc.; then

ν0,−2 = 2ν0,−1−ν0,1, ν0,−3 = 2ν0,−2−ν0,−1 = 3ν0,−1−2ν0,1, etc.. In this way we obtain

ν0,k =

k ν0,1 − (k − 1)ν0,−1, k ≥ 1

−k ν0,−1 + (k + 1)ν0,1, k ≤ −1.

The functions νk are obtained in the same way. For example, ν1(ξ) = 2ν0,1 − ν0(ξ),

ν2(ξ) = 2ν0,2 − ν1(ξ) = 2ν0,1 − 2ν0,−1 + ν0(ξ), etc.; then ν−1(ξ) = 2ν0,−1 − ν0(ξ),

ν−2(ξ) = 2ν0,−2 − ν−1(ξ) = 2ν0,−1 − 2ν0,1 + ν0(ξ), etc.. In general, we have

νk(ξ) =

2ν0,k − νk−1(ξ), k ≥ 1

2ν0,k − νk+1(ξ), k ≤ −1.


We note that for νk we have the following formulas

νk(ξ) =

k (ν0,1 − ν0,−1) + ν0(ξ), k = 2p, p ∈ Z(k + 1)ν0,1 − (k − 1)ν0,−1 − ν0(ξ), k = 2p+ 1, p ∈ Z

.

Denoting the inverse of the function νk by ξk, we dene the function

F (ν) =

ξ01, ν = ν0,k, k = 2p, p ≥ 1

ξ02, ν = ν0,k, k = 2p+ 1, p ≥ 0

ξk(ν), ν ∈ (ν0,k, ν0,k+1) , k ≥ 1

ξ02, ν = ν0,1

ξ0(ν), ν ∈ (ν0,−1, ν0,1)

ξ01, ν = ν0,−1

ξk(ν), ν ∈ (ν0,k−1, ν0,k) , k ≤ −1

ξ01, ν = ν0,k, k = 2p− 1, p ≤ 0

ξ02, ν = ν0,k, k = 2p, p ≤ −1

,

which is at least of class C3.

Remark 3.43. When C1 = C∗1 = 1, c∗1 = 0 and ξ00 = (9/4)3/2, the plots of

ν0(ξ) =

∫ ξ

ξ00

√3τ2 −

(−τ8/3 + 3C1τ2 − 3

)τ4(−τ8/3 + 3C1τ2 − 3

) dτ,

ν1(ξ) = 2ν02 − ν(ξ), h−1(ξ) = 2ν01 − ν(ξ), and of corresponding prole curves σ0(ξ) =

(1/ξ, ν0(ξ)), σ1(ξ) = (1/ξ, ν1(ξ)), and σ−1(ξ) = (1/ξ, ν−1(ξ)), for ξ ∈ (ξ01, ξ02), are

represented in Figures 3.7, 3.8, 3.9, 3.10.

Remark 3.44. The function F is periodic and its main period is 2 (ν0,1 − ν0,−1).

Remark 3.45. The function F depends on C1 and C∗1 .

We dene σk(ξ) = (χ(ξ), νk(ξ)), ξ ∈ (ξ01, ξ02), where k ∈ Z. From Theorem 3.38,

we know that (DC1 , gC1) is isometric to the surface of revolution given by

ΨC1,C∗1(ξ, θ) =

(χ(ξ) cos

θ

C∗1

, χ(ξ) sinθ

C∗1

, νk(ξ)

), (ξ, θ) ∈ DC1 .

We can reparameterize σk and one obtains

σk(ν) =

σ (ξk(ν)) = ((χ ξk)(ν), ν) = ((χ F )(ν), ν) , ν ∈ (ν0,k, ν0,k+1) , k ≥ 1

σ (ξ0(ν)) = ((χ ξ0)(ν), ν) = ((χ F )(ν), ν) , ν ∈ (ν0,−1, ν0,1) , k = 0

σ (ξk(ν)) = ((χ ξk)(ν), ν) = ((χ F )(ν), ν) , ν ∈ (ν0,k−1, ν0,k) , k ≤ −1

.

Now, let us consider the periodic curve

σ(ν) = ((χ F )(ν), ν) , ν ∈ R.


Figure 3.7: Plot of ν0. Figure 3.8: Plot of ν0, ν1 and ν−1.

Figure 3.9: Plot of σ0. Figure 3.10: Plot of σ0, σ1 and σ−1.

The curve σ is the prole curve of SC1,C∗1and it is the graph of the function χ F

depending on ν and dened on the whole Oz (or Oν). We note that σ is at least of

class C3.

Theorem 3.46. The surface of revolution given by

ΨC1,C∗1(ν, θ) =

((χ F )(ν) cos θ

C∗1

, (χ F )(ν) sin θ

C∗1

, ν

), (ν, θ) ∈ R2,

is complete and, on an open dense subset, it is locally isometric to (DC1 , gC1). The

induced metric is given by

gC1,C∗1(ν, θ) =

3F 2(ν)

3F 2(ν)− (C∗1 )

2 (−F 8/3(ν) + 3C1F 2(ν)− 3)dν2 +

1

F 2(ν)dθ2,

(ν, θ) ∈ R2. Moreover, gradK = 0 at any point of that open dense subset, and 1−K > 0

everywhere.


From Theorem 3.46 we easily get the following result.

Proposition 3.47. The universal cover of the surface of revolution given by ΨC1,C∗1

is R2 endowed with the metric gC1,C∗1. It is complete, 1 − K > 0 on R2 and, on an

open dense subset, it is locally isometric to (DC1 , gC1) and gradK = 0 at any point.

Moreover any two surfaces(R2, gC1,C∗

1

)and

(R2, gC1,C∗′

1

)are isometric.

Proof. We only have to prove the last statement. We construct the isometry between(R2, gC1,C∗

1

)and

(R2, gC1,C∗′

1

)in a natural way, in the sense that, for example, it maps

the interval (ν0,−1, ν0,1) corresponding to C∗1 onto the interval (ν0,−1, ν0,1) corresponding

to C∗′1 . Repeating this process, we obtain an (at least) C3 dieomorphism of R2. It is

easy to see that such dieomorphism is a global isometry.

The second step is to construct eectively the biconservative immersion from the

surface(R2, gC1,C∗

1

)in S3, or from SC1,C∗

1in S3. The geometric ideea of the construction

is the following: from each piece S±C1,C∗

1 ,c∗1of SC1,C∗

1we go back to (DC1 , gC1) and then,

using ϕC1 and a specic choice of+ or− and of the constant c1, we get our biconservative

immersion ΦC1,C∗1. Again, the choices of + and −, and of the constant c1 are uniquely

determined (modulo 2π, for c1) by the rst choice of +, or of −, and of the constant

c1.

Further, we will give more details about this construction. We recall that, from

Theorem 3.31 and Lemma 3.35, we have that ΦC1 : (DC1 , gC1) → S3,

ϕC1(ξ, θ) =

(√1− 1

C1ξ2cos ζ,

√1− 1

C1ξ2sin ζ,

cos(√C1θ)√

C1ξ,sin(

√C1θ)√C1ξ

),

with ζ(ξ) = ± (ζ0(ξ) + c1), is a biconservative immersion in S3 and

limξξ01

ζ0(ξ) = ζ0,−1 > −∞, limξξ02

ζ0(ξ) = ζ0,1 <∞.

In order to construct a biconservative immersion from(R2, gC1,C∗

1

)in S3, starting

with the rst component of the parametrization, we consider the following continuous

functions dened on [ξ01, ξ02]:

Φ1k(ξ) =

√

1− 1C1ξ2

cos (ζ0(ξ) + c1,k) , ξ ∈ (ξ01, ξ02)√1− 1

C1ξ201cos (ζ0,−1 + c1,k) , ξ = ξ01√

1− 1C1ξ202

cos (ζ0,1 + c1,k) , ξ = ξ02

,

where c1,k ∈ R for any k ∈ Z.Next, consider the function Φ1 : R → R dened by

Φ1(ν) =

(Φ1k F

)(ν), ν ∈ [ν0,k, ν0,k+1] , k ≥ 1(

Φ10 F

)(ν), ν ∈ [ν0,−1, ν0,1](

Φ1k F

)(ν), ν ∈ [ν0,k−1, ν0,k] , k ≤ −1

. (3.28)


We will prove that Φ1 is of class C3. Since F is a periodic function, with the main period

2 (ν0,1 − ν0,−1), it is enough to ask Φ1 to be a C3 function on the interval (ν0,−2, ν0,2) =

(2ν0,−1 − ν0,1, 2ν0,1 − ν0,−1). This means that it is enough to study the behaviour of F

at ν0,−1 and ν0,1.

First, we ask Φ1 to be continuous at ν0,−1 and ν0,1, i.e.,

limνν0,1

Φ1(ν) = limνν0,1

Φ1(ν) ∈ R, limνν0,−1

Φ1(ν) = limνν0,−1

Φ1(ν) ∈ R.

Since

limνν0,1


Φ10(F (ν)) = lim

νν0,1Φ10(ξ0(ν))

= limξξ02

Φ10(ξ) =

√1− 1

C1ξ202cos (ζ0,1 + c1,0) ∈ R

and

limνν0,1


Φ11(F (ν)) = lim

νν0,1Φ11(ξ1(ν))

= limξξ02

Φ11(ξ) =

√1− 1

C1ξ202cos (ζ0,1 + c1,1) ∈ R,

we get that cos (ζ0,1 + c1,0) = cos (ζ0,1 + c1,1). Therefore, we have two cases, as c1,1 =

c1,0 + 2s1π or c1,1 = −2ζ0,1 − c1,0 + 2s1π, where s1 ∈ Z, i.e.,

c1,1 ≡ c1,0 (mod 2π) or c1,1 ≡ (−2ζ0,1 − c1,0) (mod 2π) .

In a similar way, for ν0,−1, we have

limνν0,−1


Φ10(F (ν)) = lim

νν0,−1

Φ10(ξ0(ν))

= limξξ01

Φ10(ξ) =

√1− 1

C1ξ201cos (ζ0,−1 + c1,0) ∈ R

and

limνν0,−1


Φ11(F (ν)) = lim

νν0,−1

Φ11(ξ−1(ν))

= limξξ01

Φ11(ξ) =

√1− 1

C1ξ201cos (ζ0,−1 + c1,−1) ∈ R.

Hence, we must have cos (ζ0,−1 + c1,0) = cos (ζ0,−1 + c1,−1). Therefore we have again

two cases as c1,−1 = c1,0 +2s−1π or c1,−1 = −2ζ0,−1 − c1,0 +2s−1π, where s−1 ∈ Z, i.e.,c1,−1 ≡ c1,0 (mod 2π) or c1,−1 ≡ (−2ζ0,−1 − c1,0) (mod 2π).


Further, we ask that Φ1 to be a C1 function on the interval (ν0,−2, ν0,2). Thus we

must see what happens at ν0,−1 and ν0,1.

When ν ∈ (ν0,−1, ν0,1), we have

(Φ1)′(ν) =

(Φ1

0

)′(ξ0(ν))ξ

′0(ν) =

(Φ1

0

)′(ξ0(ν))

1

ν′0(ξ0(ν))

=

1

C1ξ30(ν)√1− 1

C1ξ20(ν)

cos (ζ0(ξ0(ν)) + c1,0)−

√1− 1

C1ξ20(ν)·

·√νξ

4/30 (ν)

(−1 + C1ξ20(ν))

√−ξ8/30 (ν) + 3C1ξ20(ν)− 3

sin (ζ0(ξ0(ν)) + c1,0)

·

·ξ20(ν)

√−ξ8/30 (ν) + 3C1ξ20(ν)− 3√

3ξ20(ν)− (C∗1 )

2(−ξ8/30 (ν) + 3C1ξ20(ν)− 3

) .

Thus

limνν0,1

(Φ1)′(ν) = lim

ξξ02

1

C1ξ3√1− 1

C1ξ2

cos (ζ0(ξ) + c1,0)−√

1− 1

C1ξ2·

·√C1ξ

4/3

(−1 + C1ξ2)√−ξ8/3 + 3C1ξ2 − 3

sin (ζ0(ξ) + c1,0)

]·

· ξ2√−ξ8/3 + 3C1ξ2 − 3√

3ξ2 − (C∗1 )

2 (−ξ8/3 + 3C1ξ2 − 3)

= −

√C1

3

(1− 1

C1ξ202

)ξ7/302

−1 + C1ξ202sin (ζ0,1 + c1,0) ∈ R

and

limνν0,−1

(Φ1)′(ν) = lim

ξξ01

1

C1ξ3√

1− 1C1ξ2

cos (ζ0(ξ) + c1,0)−√

1− 1

C1ξ2·

·√C1ξ

4/3

(−1 + C1ξ2)√

−ξ8/3 + 3C1ξ2 − 3sin (ζ0(ξ) + c1,0)

]·

· ξ2√−ξ8/3 + 3C1ξ2 − 3√

3ξ2 − (C∗1 )

2 (−ξ8/3 + 3C1ξ2 − 3)

= −

√C1

3

(1− 1

C1ξ201

)ξ7/301

−1 + C1ξ201sin (ζ0,−1 + c1,0) ∈ R.


When ν ∈ (ν0,1, ν0,2), we have

(Φ1)′(ν) =

(Φ1

1

)′(ξ1(ν))ξ

′1(ν) =

(Φ1

1

)′(ξ1(ν))

1

ν′1(ξ1(ν))

=

1

C1ξ31(ν)√1− 1

C1ξ21(ν)

cos (ζ0 (ξ1(ν)) + c1,1)−

√1− 1

C1ξ21(ν)·

·√C1ξ

4/31 (ν)

(−1 + C1ξ21(ν))

√−ξ8/31 (ν) + 3C1ξ21(ν)− 3

sin (ζ0 (ξ1(ν)) + c1,1)

·

·

−ξ21(ν)

√−ξ8/31 (ν) + 3C1ξ21(ν)− 3√

3ξ21(ν)− (C∗1 )

2(−ξ8/31 (ν) + 3C1ξ21(ν)− 3

) .

Thus

limνν0,1

(Φ1)′(ν) = − lim

ξξ02

1

C1ξ3√

1− 1C1ξ2

cos (ζ0(ξ) + c1,1)−√1− 1

C1ξ2·

·√C1ξ

4/3

(−1 + C1ξ2)√

−ξ8/3 + 3C1ξ2 − 3sin (ζ0(ξ) + c1,1)

]·

· ξ2√−ξ8/3 + 3C1ξ2 − 3√

3ξ2 − (C∗1 )

2 (−ξ8/3 + 3C1ξ2 − 3)

=

√C1

3

(1− 1

C1ξ202

)ξ7/302

−1 + C1ξ202sin (ζ0,1 + c1,1) ∈ R.

When ν ∈ (ν0,−2, ν0,−1), we have

(Φ1)′(ν) =

(Φ1

−1

)′(ξ−1(ν))ξ

′−1(ν) =

(Φ1

−1

)′(ξ−1(ν))

1

ν′−1(ξ−1(ν))

=

1

C1ξ3−1(ν)√1− 1

C1ξ2−1(ν)

(ζ0 (ξ−1(ν)) + c1,−1)−√

1− 1

C1ξ2−1(ν)·

·√C1ξ

4/3−1 (ν)(

−1 + C1ξ2−1(ν))√

−ξ8/3−1 (ν) + 3C1ξ2−1(ν)− 3sin (ζ0 (ξ−1(ν)) + c1,−1)

·

·

−ξ2−1(ν)

√−ξ8/3−1 (ν) + 3C1ξ2−1(ν)− 3√

3ξ2−1(ν)− (C∗1 )

2(−ξ8/3−1 (ν) + 3C1ξ2−1(ν)− 3

) .


Hence

limνν0,−1

(Φ1)′(ν) = − lim

ξξ01

1

C1ξ3√

1− 1C1ξ2

cos (ζ0(ξ) + c1,−1)−√

1− 1

C1ξ2·

·√C1ξ

4/3

(−1 + C1ξ2)√

−ξ8/3 + 3C1ξ2 − 3sin (ζ0(ξ) + c1,−1)

]·

· ξ2√−ξ8/3 + 3C1ξ2 − 3√

3ξ2 − (C∗1 )

2 (−ξ8/3 + 3C1ξ2 − 3)

=

√C1

3

(1− 1

C1ξ201

)ξ7/301

−1 + C1ξ201sin (ζ0,−1 + c1,−1) ∈ R.

The function Φ1 is of class C1 on (ν0,−2, ν0,2) if and only if

limνν0,1

(Φ1)′(ν) = lim

νν0,1

(Φ1)′(ν) ∈ R and lim

νν0,−1

(Φ1)′(ν) = lim

νν0,−1

(Φ1)′(ν) ∈ R.

The above equalities are equivalent to

sin (ζ0,1 + c1,0) = − sin (ζ0,1 + c1,1) and sin (ζ0,−1 + c1,0) = − sin (ζ0,−1 + c1,−1) .

We recall that, from the continuity of Φ1, there are two possibilities for each c1,1 and

c1,−1 as follows

• If c1,1 = c1,0 + 2s1π, i.e., c1,1 ≡ c1,0 (mod 2π), we get

sin (ζ0,1 + c1,1) = sin (ζ0,1 + c1,0 + 2s1π) = sin (ζ0,1 + c1,0) ;

• If c1,1 = −2ζ0,1 − c1,0 + 2s1π, i.e., c1,1 ≡ (−2ζ0,1 − c1,0) (mod 2π), we get

sin (ζ0,1 + c1,1) = sin (ζ0,1 − 2ζ0,1 − c1,0 + 2s1π) = − sin (ζ0,1 + c1,0) ;

• If c1,−1 = c1,0 + 2s−1π, i.e., c1,−1 ≡ c1,0 (mod 2π), we get

sin (ζ0,−1 + c1,−1) = sin (ζ0,−1 + c1,0 + 2s−1π) = sin (ζ0,−1 + c1,0) ;

• If c1,−1 = −2ζ0,−1 − c1,0 + 2s−1π, i.e., c1,−1 ≡ (−2ζ0,−1 − c1,0) (mod 2π), we get

sin (ζ0,−1 + c1,−1) = sin (ζ0,−1 − 2ζ0,−1 − c1,0 + 2s−1π) = − sin (ζ0,−1 + c1,0) .

Then we can choose

c1,1 ≡ (−2ζ0,1 − c1,0) (mod 2π) and c1,−1 ≡ (−2ζ0,−1 − c1,0) (mod 2π) . (3.29)

With this choice, we show that Φ1 is of class C2 on (ν0,−2, ν0,2).


The reasoning is similar to that from the proof of C1 smoothness of Φ1.

When ν ∈ (ν0,−1, ν0,1), we have(Φ1)′′

(ν) =(Φ10

)′′(ξ0(ν))

1

(ν ′0(ξ0(ν)))2 −

(Φ10

)′(ξ0(ν))

ν ′′0 (ξ0(ν))

(ν ′0(ξ0(ν)))3 .

From the expressions of Φ10 and of ν0, one can compute (Φ1)′′(ν) and we obtain

limνν0,1

(Φ1)′′

(ν) =

[− C1ξ

14/302

3 (−1 + C1ξ202)2

(1− 1

C1ξ202

)1/2

+

+− 4

3ξ2/302 + 3C1

3C1

(1− 1

C1ξ202

)−1/2]cos (ζ0,1 + c1,0) ∈ R

and

limνν0,−1

(Φ1)′′

(ν) =

[− C1ξ

14/301

3 (−1 + C1ξ201)2

(1− 1

C1ξ201

)1/2

+

+− 4

3ξ2/301 + 3C1

3C1

(1− 1

C1ξ201

)−1/2]cos (ζ0,−1 + c1,0) ∈ R.

When ν ∈ (ν0,1, ν0,2), we have(Φ1)′′

(ν) =(Φ11

)′′(ξ1(ν))

1

(ν ′1(ξ1(ν)))2 −

(Φ11

)′(ξ1(ν))

ν ′′1 (ξ1(ν))

(ν ′1(ξ1(ν)))3 ,

and, as above, one obtains

limνν0,1

(Φ1)′′

(ν) =

[− C1ξ

14/302

3 (−1 + C1ξ202)2

(1− 1

C1ξ202

)1/2

+

+− 4

3ξ2/302 + 3C1

3C1

(1− 1

C1ξ202

)−1/2]cos (ζ0,1 + c1,1) ∈ R

For ν ∈ (ν0,−2, ν0,−1), we have(Φ1)′′

(ν) =(Φ1−1

)′′(ξ−1(ν))

1(ν ′−1(ξ−1(ν))

)2 −(Φ1−1

)′(ξ−1(ν)) ·

ν ′′−1(ξ−1(ν))(ν ′−1(ξ−1(ν))

)3and one gets

limνν0,−1

(Φ1)′′

(ν) =

[− C1ξ

14/301

3 (−1 + C1ξ201)2

(1− 1

C1ξ201

)1/2

+

+− 4

3ξ2/301 + 3C1

3C1

(1− 1

C1ξ201

)−1/2]cos (ζ0,−1 + c1,−1) ∈ R

From (3.29) one obtains

limνν0,1

(Φ1)′′

(ν) = limνν0,1

(Φ1)′′

(ν) ∈ R, limνν0,−1

(Φ1)′′

(ν) = limνν0,−1

(Φ1)′′

(ν) ∈ R,


and consequently, for this choice of c1,1 and c1,−1, we get that Φ1 is of class C2 on

(h0,−2, h0,2).

Using the same method, one can prove that Φ1 is of class C3 on (ν0,−2, ν0,2). More

precisely, one sees that

limνν0,1

(Φ1)′′′

(ν) = limνν0,1

(Φ1)′′′

(ν) =

=

√C1

(1− 1

C1ξ202

)ξ13/302

(6− 6C1 (C

∗1 )

2+ 8

3 (C∗1 )

2ξ2/302

)4 · 35/2(−1 + C1ξ202)

·

·(6C1 −

8

3ξ2/302

)+C

3/21 ξ702

√1− 1

C1ξ202

33/2 (−1 + C1ξ202)3 −

ξ7/302

(6C1 − 8

3ξ2/302

)2√3C1

√1− 1

C1ξ202

−

−5√C1ξ

13/302

(6C1 − 8

3ξ2/302

)√1− 1

C1ξ202

35/2 (−1 + C1ξ202)+

+C

3/21 ξ

19/302

(6C1 − 8

3ξ2/302

)√1− 1

C1ξ202

33/2 (−1 + C1ξ202)2

sin (ζ0,1 + c1,0) ∈ R

and

limνν0,−1

(Φ1)′′′


(Φ1)′′′

(ν) =

=

√C1

(1− 1

C1ξ201

)ξ13/301

(6− 6C1 (C

∗1 )

2+ 8

3 (C∗1 )

2ξ2/301

)4 · 35/2(−1 + C1ξ201)

·

·(6C1 −

8

3ξ2/301

)+C

3/21 ξ701

√1− 1

C1ξ201

33/2 (−1 + C1ξ201)3 −

ξ7/301

(6C1 − 8

3ξ2/301

)2√3C1

√1− 1

C1ξ201

−

−5√C1ξ

13/301

(6C1 − 8

3ξ2/301

)√1− 1

C1ξ201

35/2 (−1 + C1ξ201)+

+C

3/21 ξ

19/301

(6C1 − 8

3ξ2/301

)√1− 1

C1ξ201

33/2 (−1 + C1ξ201)2

sin (ζ0,−1 + c1,0) ∈ R.

In general, if we ask Φ1 to be of class C3 on R, since F is a periodic function, it can be

shown that we have the following relations between two consecutive c1,k, where k ∈ Z:

c1,k ≡

(−2ζ0,1 − c1,k−1) (mod 2π) , k = 2p+ 1, p ∈ N(−2ζ0,−1 − c1,k−1) (mod 2π) , k = 2p, p ∈ N(−2ζ0,−1 − c1,k+1) (mod 2π) , k = 2p− 1, p ∈ Z−

(−2ζ0,1 − c1,k+1) (mod 2π) , k = 2p, p ∈ Z−

, (3.30)


or, equivalently,

c1,k ≡

(−2ζ0,1 − c1,k−1) (mod 2π) k = 2p+ 1, p ∈ Z(−2ζ0,−1 − c1,k−1) (mod 2π) k = 2p, p ∈ Z

.

We note that for c1,k, we also have the following formulas

c1,k ≡

(k (ζ0,1 − ζ0,−1) + c1,0) (mod 2π) , k = 2p, p ∈ Z((k − 1)ζ0,−1 − (k + 1)ζ0,1 − c1,0) (mod 2π) , k = 2p+ 1, p ∈ Z

. (3.31)

To study the second component of the parametrization ϕC1 , we work in a similar

way as for the rst one. We consider the following continuous functions dened on

[ξ01, ξ02]:

Φ2k(ξ) =

(−1)k

√1− 1

C1ξ2sin (ζ0(ξ) + c1,k) , ξ ∈ (ξ01, ξ02)

(−1)k√1− 1

C1ξ201sin (ζ0,−1 + c1,k) , ξ = ξ01

(−1)k√

1− 1C1ξ202

sin (ζ0,1 + c1,k) , ξ = ξ02

,

where c1,k ∈ R, for any k ∈ Z, are given by (3.30).

Then, we consider the function Φ2 : R → R dened by

Φ2(ν) =

(Φ2k F

)(ν), ν ∈ [ν0,k, ν0,k+1] , k ≥ 1(

Φ20 F

)(ν), ν ∈ [ν0,−1, ν0,1](

Φ2k F

)(ν), ν ∈ [ν0,k−1, ν0,k] , k ≤ −1

. (3.32)

We show that, with these choices of the constants c1,k, Φ2 is of class C3. The proof

is similar to the proof of C3 smoothness of Φ1, as we will see further.

Since F is a periodic function, with main period 2 (ν0,1 − ν0,−1), it is enough to

prove that Φ2 is of class C3 on (ν0,−2, ν0,2). This means that it is enough to study what

happens at ν0,−1 and ν0,1.

Since sin (ζ0,−1 + c1,−1) = − sin (ζ0,−1 + c1,0) and sin (ζ0,1 + c1,1) = − sin (ζ0,1 + c1,0),

it is easy to see that Φ2 is continuous at ν0,−1 and ν0,1, as

limνν0,1


Φ2(ν) =

√1− 1

C1ξ202sin (ζ0,1 + c1,0) ∈ R

and

limνν0,−1


Φ2(ν) =

√1− 1

C1ξ201sin (ζ0,−1 + c1,0) ∈ R.

Therefore, we get that Φ2 is continuous on (ν0,−2, ν0,2).

Since cos (ζ0,−1 + c1,−1) = cos (ζ0,−1 + c1,0) and cos (ζ0,1 + c1,1) = cos (ζ0,1 + c1,0),

we obtain that

limνν0,1

(Φ2)′(ν) = lim

νν0,1

(Φ2)′(ν) =

√C1

3

(1− 1

C1ξ202

)ξ7/302

−1 + C1ξ202cos (ζ0,1 + c1,0) ∈ R


and

limνν0,−1

(Φ2)′(ν) = lim

νν0,−1

(Φ2)′(ν) =

√C1

3

(1− 1

C1ξ201

)ξ7/301

−1 + C1ξ201cos (ζ0,−1 + c1,0) ∈ R.

Therefore, we get that Φ2 is of class C1 on (ν0,−2, ν0,2).

Next, from sin (ζ0,−1 + c1,−1) = − sin (ζ0,−1 + c1,0) and sin (ζ0,1 + c1,1) = − sin (ζ0,1 + c1,0),

we get that

limνν0,1

(Φ2)′′

(ν) = limνν0,1

(Φ2)′′

(ν) =

=

[− C1ξ

14/302

3 (−1 + C1ξ202)2

(1− 1

C1ξ202

)1/2

+

+− 4

3ξ2/302 + 3C1

3C1

(1− 1

C1ξ202

)−1/2]sin (ζ0,1 + c1,0) ∈ R

and

limνν0,−1

(Φ2)′′


(Φ2)′′

(ν) =[− C1ξ

14/301

3 (−1 + C1ξ201)2

(1− 1

C1ξ201

)1/2

+

+− 4

3ξ2/301 + 3C1

3C1

(1− 1

C1ξ201

)−1/2]sin (ζ0,−1 + c1,0) ∈ R.

Therefore, we have that Φ2 is of class C2 on (ν0,−2, ν0,2).

Similarly, for the third derivative of Φ2, using the relations cos (ζ0,−1 + c1,−1) =

cos (ζ0,−1 + c1,0) and cos (ζ0,1 + c1,1) = cos (ζ0,1 + c1,0), one obtains

limνν0,1

(Φ2)′′′

(ν) = limνν0,1

(Φ2)′′′

(ν) =

= −

√C1

(1− 1

C1ξ202

)ξ13/302

(6− 6C1 (C

∗1 )

2+ 8

3 (C∗1 )

2ξ2/302

)4 · 35/2(−1 + C1ξ202)

·

·(6C1 −

8

3ξ2/302

)+C

3/21 ξ702

√1− 1

C1ξ202

33/2 (−1 + C1ξ202)3 −

ξ7/302

(6C1 − 8

3ξ2/302

)2√3C1

√1− 1

C1ξ202

−

−5√C1ξ

13/302

(6C1 − 8

3ξ2/302

)√1− 1

C1ξ202

35/2 (−1 + C1ξ202)+

+C

3/21 ξ

19/302

(6C1 − 8

3ξ2/302

)√1− 1

C1ξ202

33/2 (−1 + C1ξ202)2

cos (ζ0,1 + c0) ∈ R


and

limνν0,−1

(Φ2)′′′


(Φ2)′′′

(ν) =

= −

√C1

(1− 1

C1ξ201

)ξ13/301

(6− 6C1 (C

∗1 )

2+ 8

3 (C∗1 )

2ξ2/301

)4 · 35/2(−1 + C1ξ201)

·

·(6C1 −

8

3ξ2/301

)+C

3/21 ξ701

√1− 1

C1ξ201

33/2 (−1 + C1ξ201)3 −

ξ7/301

(6C1 − 8

3ξ2/301

)2√3C1

√1− 1

C1ξ201

−

−5√C1ξ

13/301

(6C1 − 8

3ξ2/301

)√1− 1

C1ξ201

35/2 (−1 + C1ξ201)+

+C

3/21 ξ

19/301

(6C1 − 8

3ξ2/301

)√1− 1

C1ξ201

33/2 (−1 + C1ξ201)2

cos (ζ0,−1 + c1,0) ∈ R.

Therefore, we see that Φ2 is of class C3 on (ν0,−2, ν0,2).

Since F is periodic, one can prove that Φ2 is of class C3 on whole R.For the third component of the parametrization ϕC1 , we consider the following func-

tion

Φ30(ξ) =

1√C1ξ

, ξ ∈ [ξ01, ξ02] ,

It is obvious that Φ30 is a smooth function on [ξ01, ξ02].

Let us consider a new function Φ3 : R → R dened by

Φ3(ν) = (Φ30 F )(ν), ν ∈ R. (3.33)

Since F is at least of class C3 on R and Φ30 is smooth on [ξ01, ξ02], it follows that Φ

3 is

at least of class C3 on R.For the fourth component of the parametrization ϕC1 , we dene Φ4 as Φ3, i.e.,

Φ4(ν) = (Φ40 F )(ν), ν ∈ R, (3.34)

where Φ40(ξ) = 1/

(√C1ξ

), for any ξ ∈ [ξ01, ξ02].

Now, we can conclude with the following theorem.

Theorem 3.48. The map ΦC1,C∗1:(R2, gC1,C∗

1

)→ S3, dened by

ΦC1,C∗1(ν, θ) = ϕC1(F (ν), θ) =

(Φ1(ν),Φ2(ν),Φ3(ν) cos(

√C1θ),Φ

4(ν) sin(√C1θ)

),

(ν, θ) ∈ R2, where Φ1, Φ2, Φ3 and Φ4 are given by (3.28), (3.32), (3.33) and (3.34),

respectively, and the constants c1,k are given by (3.31), is a biconservative immersion.


Proof. Obviously, for ν ∈ (ν0,k, ν0,k+1), when k ≥ 1, or ν ∈ (ν0,−1, ν0,1), or ν ∈(ν0,k−1, ν0,k), when k ≤ −1, ΦC1,C∗

1is an isometric immersion and it is biconservat-

ive. As ΦC1,C∗1is a map of class C3 and the biconservative equation is a third-degree

equation, by continuity, we get that ΦC1,C∗1is biconservative on R2.

Remark 3.49. We note that ΦC1,C∗1has self-intersections (along circles).

We end this chapter by verifying (partially) the correctness of the above construc-

tion.

More precisely, if we denote by S+C1,c1,0

the image of ΦC1,C∗1((ν0,−1, ν0,1)× R), we

can see that its boundary in R4 is given by the curves:(√1− 1

C1ξ201cos (ζ0,−1 + c1,0) ,

√1− 1

C1ξ201sin (ζ0,−1 + c1,0) ,

cos(√C1θ

)√C1ξ01

,sin(√C1θ

)√C1ξ01

)

and(√1− 1

C1ξ202cos (ζ0,1 + c1,0) ,

√1− 1

C1ξ202sin (ζ0,1 + c1,0) ,

cos(√C1θ

)√C1ξ02

,sin(√C1θ

)√C1ξ02

).

These curves are two circles in the ane planes(√1− 1

C1ξ201cos (ζ0,−1 + c1,0) ,

√1− 1

C1ξ201sin (ζ0,−1 + c1,0) , 0, 0

)+ span e3, e4

and(√1− 1

C1ξ202cos (ζ0,1 + c1,0) ,

√1− 1

C1ξ202sin (ζ0,1 + c1,0) , 0, 0

)+ span e3, e4 ,

respectively. The radii of the lower circle is 1/(√C1ξ01

)and of the upper one

1/(√C1ξ02

), respectively.

Now if we consider S−C1,c1,1

the image of ΦC1,C∗1((ν0,1, ν0,2)× R), we can see that its

boundary in R4 is given by the curves:(√1− 1

C1ξ202cos (ζ0,1 + c1,1) ,−

√1− 1

C1ξ202sin (ζ0,1 + c1,1) ,

cos(√C1θ

)√C1ξ02

,sin(√C1θ

)√C1ξ02

)

and(√1− 1

C1ξ201cos (ζ0,−1 + c1,1) ,−

√1− 1

C1ξ201sin (ζ0,−1 + c1,1) ,

cos(√C1θ

)√C1ξ01

,sin(√C1θ

)√C1ξ01

).

These curves are two circles and the radii of them are 1/(√C1ξ02

)(for the lower

circle) and 1/(√C1ξ01

)(for the upper circle).

Since c1,1 ≡ (−2ζ0,1 − c1,0) (mod 2π), it is easy to see that the lower circle from

the boundary of S−C1,c1,1

coincide with the upper circle from the boundary of S+C1,c1,0

.


Then, at a common boundary point, we get that the tangent plane to the closure in R4,

S+C1,c1,0

, of S+C1,c1,0

is spanned by a vector tangent to the boundary and the vector− ξ4/302√

3(C1ξ202 − 1

) sin (ζ0,1 + c1,0) ,ξ4/302√

3(C1ξ202 − 1

) cos (ζ0,1 + c1,0) , 0, 0

.

At the same boundary point, the tangent plane to S−C1,c1,1

is spanned by a vector tangent

to the boundary and the vector ξ4/302√

3(C1ξ202 − 1

) sin (ζ0,1 + c1,1) ,ξ4/302√

3(C1ξ202 − 1

) cos (ζ0,1 + c1,1) , 0, 0

.

As c1,1 ≡ (−2ζ0,1 − c1,0) (mod 2π), the two tangent planes coincide.

However, we must then check that we have a C3 smooth gluing.

Further, we consider C1 = C∗1 = 1, c∗1 = 0 and we start with + in the expression of

ν.

The construction of a complete biconservative surface in S3 can be summarized in

the diagram from Figure 3.11.

Some numerical experiments suggest that ΦC1,C∗1is not periodic and it has self-

intersections along circles parallel to Ox3x4.

The projection of ΦC1,C∗1on the Ox1x2 plane is a curve which lies in the annulus

of radii√1− 1/

(C1ξ201

)and

√1− 1/

(C1ξ202

). It has self-intersections and is dense in

the annulus. This is illustrated in Figure 3.12.

In Figure 3.13, we represent the signed curvature of the prole curve of SC1,C∗1, and

it suggests the fact that we have a smooth gluing.

The signed curvature of the curve obtained projecting ΦC1,C∗1on the Ox1x2 plane

is represented in Figure 3.14 and it suggests the fact that ΦC1,C∗1is at least of class C3.

However, we can state the following open problem.

Open problem. Does there exist a non-CMC biconservative immersion ΦC1,C∗1that

is double periodic, thus providing a compact non-CMC biconservative surface in S3?

Concerning the biharmonic surfaces in S3 we have the following classication result.

Theorem 3.50 ([13]). Let φ : M2 → S3 be a proper-biharmonic surface. Then φ(M)

is an open part of the small hypersphere S2(1/√2).

Figure 3.11: The idea of the construction of complete biconservative surfaces in S3.


Figure 3.12: The projection of ΦC1,C∗1on the Ox1x2 plane.

Figure 3.13: The signed curvature of the prole curve of SC1,C∗1.

Figure 3.14: The signed curvature of the curve obtained projecting ΦC1,C∗1on the Ox1x2

plane.

Chapter 4Biconservative surfaces in

arbitrary Riemannian manifolds

In this chapter we present general properties of biconservative surfaces in arbitrary

Riemannian manifolds. We nd the link between biconservativity, the property of the

shape operator AH to be a Codazzi tensor eld, the holomorphicity of a generalized Hopf

function and the quality of the surface to have constant mean curvature. Then, we give

a description of the metric and of the shape operator AH for a CMC biconservative

surface. Finally, we nd a Simons type formula for biconservative surfaces and then use

it to study their geometry.

The major part of the results from this chapter is original and it can be also found

in [55]. Some of them are known results, but obtained in a dierent way.

4.1 More characterizations of biconservative sub-

manifolds

In this section we characterize biconservative submanifolds satisfying some additional

geometric hypotheses.

We begin with a study on the basic properties of submanifolds with AH parallel,

as they are the simplest biconservative submanifolds. First, we dene the principal

curvatures of a submanifold Mm of Nn as being the eigenvalue functions of AH .

Proposition 4.1. Let φ :Mm → Nn be a submanifold and λ1 ≥ · · · ≥ λm the principal

curvatures of M . If ∇AH = 0, then


(ii) λi are constant functions on M (in particular M is CMC);

107

108 Chapter 4. Biconservative surfaces in arbitrary Riemannian manifolds

(iii) A∇⊥XH

(Y )−A∇⊥YH

(X) =(RN (X,Y )H

)⊤, for any X,Y ∈ C(TM);

(iv) traceA∇⊥· H

(·) = − trace(RN (·,H)·

)⊤.

Proof. For the sake of completeness, we give the proof of the second item. More pre-

cisely, we show that λi are constant functions on M . Let us consider an arbitrary

point p ∈ M . Since AH(p) is symmetric, then AH(p) is diagonalizable. We denote by

λ1,p ≥ · · · ≥ λm,p the eigenvalues of AH(p) and then dene the continuous functions

λi :M → R, λi(p) = λi,p, for any p ∈M and any i = 1,m.

Further, we consider eii=1,m an orthonormal basis in TpM which diagonalizes

AH(p), i.e., (AH(p)) (ei) = λi(p)ei, for any i = 1,m.

Consider q ∈ M , q = p, and γ : [a, b] → M a smooth curve such that γ(a) = p and

γ(b) = q. We dene the vector elds Xi = Xi(t), along γ, such that DXidt (t) = 0, for

any t and Xi(a) = ei. It is easy to see that W (t) = (AH(γ(t))) (Xi(t)) is also a vector

eld along γ and

DW

dt(t) =

(∇γ′(t)AH

)(Xi(t)) +AH

(DXi

dt(t)

)= 0.

Now, since D(λi(p)Xi)dt (t) = 0, we get that W (t) and λi(p)Xi are parallel along γ. Since

for t = a they are equal, it follows that they coincide for any t, and in particular, for

t = b. Therefore, λi(p) are eigenvalues of AH(q). As q was chosen in an arbitrary way,

we get that λi are constant functions on M , for any i = 1,m.

We will nd, later in this chapter, some converse results of this proposition. More

precisely in the case when m = 2, we will show that, under some standard hypotheses,

a biconservative surface satises ∇AH = 0.

Corollary 4.2. Let φ : Mm → Nn(c) be a submanifold with ∇AH = 0 and c ∈ R.Then

(i) A∇⊥XH

(Y ) = A∇⊥YH

(X), for any X,Y ∈ C(TM);

(ii) traceA∇⊥· H

(·) = 0.

In the particular case of surfaces, we have a stronger result.

Proposition 4.3. Let φ : M2 → Nn be a surface. If ∇AH = 0, then M is pseudoum-

bilical or at.

Proof. Let λ1 ≥ λ2 be the principal curvatures ofM . Since ∇AH = 0, from Proposition

4.1, we have that λ1 and λ2 are constant functions on M .

4.1. More characterizations of biconservative submanifolds 109

If λ1 = λ2, obviously, M is pseudoumbilical.

If λ1 > λ2, around any point of M we can consider a local orthonormal frame eld

X1, X2 which diagonalizes AH , i.e, AH (Xi) = λiXi, for i ∈ 1, 2.Using the Ricci formula we get

R(X,Y )AH(Z) = AH(R(X,Y )Z),

for any X,Y, Z ∈ C(TM) and then

λiR (X1, X2)Xi =AH (R (X1, X2)Xi)

=AH (R(X1, X2, X1, Xi)X1 +R(X1, X2, X2, Xi)X2)

=λ1R(X1, X2, X1, Xi)X1 + λ2R(X1, X2, X2, Xi)X2,

for i ∈ 1, 2.For both i's, we obtain K = 0 on U , where K is the Gaussian curvature of M given

by K = R (X1, X2, X1, X2).

If (Mm, g) is a Riemannian manifold and T is a parallel symmetric tensor eld of

type (1, 1), then its eigenvalue functions λ1 ≥ · · · ≥ λm are constant functions on M

and, obviously, T is a Codazzi tensor eld. If m = 2, the converse result also holds.


)be a surface and consider T a symmetric tensor eld of

type (1, 1). Let λ1 ≥ λ2 be the eigenvalue functions of T . If λ1 and λ2 are constant

functions on M and T is a Codazzi tensor eld, then ∇T = 0. Moreover, if λ1 > λ2,

then(M2, g

)is at.

Proof. If λ1 = λ2 = λ, it follows that T = λI and ∇T = 0.

If λ1 > λ2, around any point of M we can consider a local orthonormal frame eld

X1, X2 which diagonalizes T . By a direct computation, one obtains(∇XjT

)(Xi) + T

(∇XjXi

)= λi∇XjXi, (4.1)

for i, j ∈ 1, 2.Since T is a Codazzi tensor eld, for appropriate choices of i and j in (4.1), we get

∇XiXj = 0, for any i, j ∈ 1, 2. It follows that K vanishes at any point of M and

∇T = 0 on M .

When T = AH we have the next result.

Corollary 4.5. Let φ :M2 → Nn be a surface and λ1 ≥ λ2 be the principal curvatures

of M . If λ1 and λ2 are constant functions on M and AH is a Codazzi tensor eld, then

∇AH = 0.


Remark 4.6. Let φ :Mm → Nn be a submanifold with the principal curvatures being

constant functions. Then M is a CMC submanifold.

We note that, in general, AH is not a Codazzi tensor eld. In the following we study

the properties of submanifolds with AH a Codazzi tensor eld, and their connection with

biconservativity, as this is the next natural step after the case when AH is parallel.

We begin with a result which follows easily from the Codazzi equation, equation

(1.9) and Proposition 1.32.

Proposition 4.7. Let φ :Mm → Nn be a submanifold. If AH is a Codazzi tensor eld,

then

(i) A∇⊥XH

(Y )−A∇⊥YH

(X) =(RN (X,Y )H

)⊤, for any X,Y ∈ C(TM);

(ii) trace∇AH = m grad(|H|2

);

(iii) traceA∇⊥· H

(·) = m2 grad

(|H|2

)− trace

(RN (·,H)·

)⊤;

(iv) M is biconservative if and only |H| is constant.

If N is an n-dimensional space form, Proposition 4.7 gives the following corollary.

Corollary 4.8. Let φ : Mm → Nn(c) be a submanifold. If AH is a Codazzi tensor

eld, then

(i) A∇⊥XH

(Y )−A∇⊥YH

(X) = 0, for any X,Y ∈ C(TM);

(ii) trace∇AH = m grad(|H|2

);


(·) = m2 grad

(|H|2

);

(iv) M is biconservative if and only if |H| is constant.

Next, we have a similar result to Proposition 4.7, where, rather than assuming AH

to be a Codazzi tensor eld, we assume that |H| = 0 at any point and Aη is a Codazzi

tensor eld, where η = H/|H| ∈ C(NM).

Proposition 4.9. Let φ : Mm → Nn be a submanifold. Assume |H| = 0 on M and

denote η = H/|H|. If Aη is a Codazzi tensor eld, then

(i) A∇⊥XH

(Y )−A∇⊥YH

(X) = X(log |H|)AH(Y )−Y (log |H|)AH(X)+(RN (X,Y )H

)⊤,

for any X,Y ∈ C(TM);

(ii) trace∇AH = AH(grad(log |H|)) +m|H| grad |H|;


(·) = AH (grad(log |H|))− trace(RN (·,H)·

)⊤;

4.2. Properties of biconservative surfaces 111

(iv) M is biconservative if and only if

AH(grad |H|) = −m2|H|2 grad |H|.

Corollary 4.10. Let φ : Mm → Nn(c) be a submanifold. Assume that |H| = 0 on M

and denote η = H/|H|. If Aη is a Codazzi tensor eld, then

(i) A∇⊥XH

(Y )− A∇⊥YH

(X) = X(log |H|)AH(Y )− Y (log |H|)AH(X), for any X,Y ∈C(TM);

(ii) trace∇AH = AH(grad(log |H|)) +m|H| grad |H|;


(·) = AH (grad(log |H|));

(iv) M is biconservative if and only if

AH(grad |H|) = −m2|H|2 grad |H|.

To end the section consider those submanifolds in space forms having H parallel

in the normal bundle (PMC submanifolds). Using the Codazzi equation, we get the

following result.

Proposition 4.11. Let φ :Mm → Nn(c) be a PMC submanifold. Then


(ii) AH is a Codazzi tensor eld;

(iii) ⟨(∇AH) (·, ·), ·⟩ is totally symmetric;

(iv) trace∇AH = 0.

4.2 Properties of biconservative surfaces

In this section we nd some of the main properties of biconservative surfaces.

One of the main results is Theorem 4.19. Before stating it, we have a lemma

which holds for an arbitrary symmetric tensor eld T of type (1, 1), then present some

properties satised when div T = 0, and nally bring all these together in Theorem

4.18.

Lemma 4.12. Let(M2, g

)be a surface and let T be a symmetric tensor eld of type

(1, 1). We have


(i)

div T = grad t− ⟨Z12, X2⟩X1 + ⟨Z12, X1⟩X2,

where t = traceT , X1, X2 is a local orthonormal frame eld on M and

Z12 = (∇X1T ) (X2)− (∇X2T ) (X1) ;

(ii) ⟨T (∂z), ∂z⟩ is holomorphic if and only if grad t = 2div T ;

(iii) ⟨T (∂z), ∂z⟩ is holomorphic if and only if

grad t = 2⟨Z12, X2⟩X1 − 2⟨Z12, X1⟩X2.

Proof. Since the rst and the third statements of the theorem follow by standard com-

putation, we will only give the proof of the second item.

SinceM is oriented, the metric g can be written locally as g = e2ρ(du2 + dv2

), where

(u, v) are positively oriented local coordinates and ρ = ρ(u, v) is a smooth function. As

usual, we denote

∂z =1

2(∂u − i∂v) and ∂z =

1

2(∂u + i∂v) .

Therefore, ⟨T (∂z), ∂z⟩ is holomorphic if and only if ∂z ⟨T (∂z), ∂z⟩ = 0. We can see that

the Christoel symbols are given by

Γ212 = Γ1

11 = −Γ122 = ρu,

where ρu = ∂ρ∂u and

Γ112 = Γ2

22 = −Γ211 = ρv.

Thus, we obtain

∇∂u∂v =∇∂v∂u = Γ112∂u + Γ2

12∂v = ρv∂u + ρu∂v,

∇∂u∂u =Γ111∂u + Γ2

11∂v = ρu∂u − ρv∂v,

∇∂v∂v =Γ122∂u + Γ2

22∂v = −ρu∂u + ρv∂v.

After some straightforward computations, we get

∂z ⟨T (∂z), ∂z⟩ =e2ρ

8(−tu + 2⟨div T, ∂u⟩+ i (tv − 2⟨div T, ∂v⟩)) ,

where t = traceT . Now, it is easy to see that ⟨T (∂z), ∂z⟩ is holomorphic if and only if

grad t = 2div T .


Remark 4.13. We note that ⟨Z12, X2⟩X1 − ⟨Z12, X1⟩X2 does not depend on the local

orthonormal frame eld X1, X2. Thus, there exists a unique global vector eld Z

such that for any local orthonormal frame eld X1, X2, on its domain of denition,

we have

Z = ⟨Z12, X2⟩X1 − ⟨Z12, X1⟩X2.

Therefore, we obtain the global formula

div T = grad t− Z.

Lemma 4.12 is the key ingredient to prove the following four propositions. First,

let(M2, g

)be a surface and consider T a symmetric tensor eld of type (1, 1) with

t = traceT .

Proposition 4.14. If t is constant, then the following relations are equivalent

(i) T is a Codazzi tensor eld;

(ii) ⟨T (∂z) , ∂z⟩ is holomorphic;

(iii) div T = 0.

Proposition 4.15. If div T = 0, then the following relations are equivalent



(iii) t is constant.

Proposition 4.16. If ⟨T (∂z) , ∂z⟩ is holomorphic, then the following relations are equi-

valent


(ii) t is constant;

(iii) div T = 0.

Proposition 4.17. If T is a Codazzi tensor eld, then the following relations are equi-

valent

(i) t is constant;


(iii) div T = 0.


Summarizing, we can now state the next theorem.


)be a surface and consider T be a symmetric tensor eld

of type (1, 1). Then any two of the following relations imply each of the others

(i) div T = 0;

(ii) t is constant;

(iii) ⟨T (∂z) , ∂z⟩ is holomorphic;

(iv) T is a Codazzi tensor eld.

Considering T = S2 or T = AH , we get the following result.

Theorem 4.19. Let φ :M2 → Nn be a surface. Then any two of the following relations

imply each of the others


(ii) |H| is constant;

(iii) ⟨AH (∂z) , ∂z⟩ is holomorphic;

(iv) AH is a Codazzi tensor eld.

Proof. For the sake of completeness, we present a sketch the proof.

First, we recall that S2 = −2|H|2I +4AH , traceS2 = 4|H|2, traceAH = 2|H|2, and

divS2 = −2 grad(|H|2

)+ 4divAH . (4.2)

It is then easy to see that

⟨S2 (∂z) , ∂z⟩ = 4⟨AH (∂z) , ∂z⟩,

and, therefore, ⟨AH (∂z) , ∂z⟩ is holomorphic if and only if ⟨S2 (∂z) , ∂z⟩ is holomorphic.

The idea of the proof is to choose a condition and then prove the equivalence between

each two other conditions using, in principal, Theorem 4.18 with T = AH or T = S2.

For example, assume that (i) holds. To prove that (ii) implies (iii) and (iv) we note

that since divS2 = 0 and |H| is constant, from (4.2), we get divAH = 0. Now, from

Theorem 4.18 with T = AH , we get (iii) and (iv). Conversely, from Proposition 4.7, we

have that hypotheses (i) and (iv) imply (ii). Since ⟨AH (∂z) , ∂z⟩ is holomorphic if and

only if ⟨S2 (∂z) , ∂z⟩ is holomorphic, from Theorem 4.18 with T = S2, it follows that (i)

and (iii) imply (ii). Using the above equivalences that we have already proved, it is

easy to see that (iii) is equivalent to (iv).

The other cases follow easily in a similar way.


Remark 4.20. We note that some of the implications in Theorem 4.19 was also ob-

tained in [50], and the fact that ⟨AH (∂z) , ∂z⟩ is holomorphic if and only if |H| isconstant was also proved in [43] under the hypothesis of biharmonicity.

Remark 4.21. If φ :M2 → Nn is a non-pseudoumbilical CMC biconservative surface,

then the set of pseudoumbilical points has no accumulation points. Also, we deduce

that if M2 is a CMC biconservative surface and is a topological sphere, then it is

pseudoumbilical (see [50]); this should be compared with the classical result: a PMC

surface M2 of genius 0 in a space form is pseudoumbilical (see [35]).

Using Corollary 4.5, Remark 4.6 and Theorem 4.19, one obtains the next result.

Theorem 4.22. Let φ : M2 → Nn be a biconservative surface. We denote by λ1 and

λ2 the principal curvatures of M . If λ1 and λ2 are constant functions on M , then

∇AH = 0.

Remark 4.23. If we replace the hypothesis M is a biconservative surface in Theorem

4.22 by ⟨AH (∂z) , ∂z⟩ is a holomorphic function the conclusion still holds.

Since any CMC surface in a 3-dimensional space form is biconservative, using The-

orem 4.19, we easily get the following well-known properties of CMC surfaces.

Corollary 4.24. Let φ :M2 → N3(c), c ∈ R, be a CMC surface. Then


(ii) AH is a Codazzi tensor eld;

(iii) ⟨AH (∂z) , ∂z⟩ is holomorphic.

For a PMC surface in an arbitrary manifold, AH is not necessarily a Codazzi tensor

eld, but when the surface is biconservative, this does happen.

Corollary 4.25. Let φ : M2 → Nn be a PMC biconservative surface. Then AH is a

Codazzi tensor eld and(RN (X,Y )H

)⊤= 0 for any X,Y ∈ C(TM).

Proof. First, we note that ∇⊥H = 0 implies |H| constant, and, if M is biconservative,

from Theorem 4.19, we have that AH is a Codazzi tensor eld.

Now, to show that(RN (X,Y )H

)⊤= 0 we only have to replace ∇⊥H = 0 in the

Codazzi equation and use the fact that AH is a Codazzi tensor eld.

The next theorem gives a description in terms of |H| and the principal curvatures

of M of the metric and of the shape operator in the direction of H for a CMC bicon-

servative surface in an arbitrary manifold.


Theorem 4.26. Let φ :M2 → Nn be a CMC biconservative surface. Denote by λ1 and

λ2 the principal curvatures of M and by µ = λ1−λ2 their dierence. Then, around any

non-pseudoumbilical point p there exists a local chart (U ;u, v) which is both isothermal

and a line of curvature coordinate system for AH . Moreover, on U , we have

⟨·, ·⟩ = 1

µ⟨·, ·⟩0,

and AH is given by

⟨AH(·), ·⟩ =(|H|2

µ+

1

2

)du2 +

(|H|2

µ− 1

2

)dv2,

or, equivalently, by

AH =

(|H|2

µ+

1

2

)du⊗ ∂u +

(|H|2

µ− 1

2

)dv ⊗ ∂v,

where ⟨·, ·⟩0 is the Euclidean metric on R2.

In the particular case when n = 3, we obtain that µ satises

µ0

∆ µ+

∣∣∣∣ 0

grad µ

∣∣∣∣20

+ 2µ

(KN + |H|2 − µ2

4|H|2

)= 0, (4.3)

where KN is the sectional curvature of N3 along M2.

Proof. Let λ1 and λ2 be the principal curvatures of M and p a non-pseudoumbilical

point in M . Then λ1, λ2 are continuous, and there exists an open neighborhood U

around p such that they are smooth functions on U , and µ = λ1 − λ2 is a positive

smooth function on U .

Let X1, X2 be a local orthonormal frame eld on U such that AH (Xi) = λiXi,

for any i ∈ 1, 2. Further, consider the connection forms on U , dened by

∇X1 = ω21X2 and ∇X2 = ω1

2X1.

Obviously, ω21 = −ω1

2. From Theorem 4.19 one obtains that AH is a Codazzi tensor

eld, i.e., on U , we have

(∇AH) (X1, X2) = (∇AH) (X2, X1) .

Using the denition of ωji , we get, on U ,

ω21 (X1) =

1

µX2 (λ1) and ω2

1 (X2) =1

µX1 (λ2) .

Now, since | traceB| = 2|H| is a constant, it is easy to see that X2 (λ1) = (X2(µ)) /2

and X1 (λ2) = − (X1(µ)) /2. If we denote byω1, ω2

the local orthornormal coframe

eld on U dual to X1, X2, one gets

ω21 =

1

2

((X2(logµ))ω

1 − (X1(logµ))ω2).


After some straightforward computations, we obtain[X1√µ,X2√µ

]= 0,

and, therefore, on U there exist coordinate functions u and v such that ∂u = X1/√µ

and ∂v = X2/√µ. Moreover the expression of the metric in isothermal coordinates on

U is

⟨·, ·⟩ = 1

µ

(du2 + dv2

)=

1

µ⟨·, ·⟩0.

Since λ1 and λ2 are principal curvatures of M , it is easy to see that

⟨AH(·), ·⟩ =1

µ

(λ1du

2 + λ2dv2).

Using λ1 + λ2 = 2|H|2 and λ1 − λ2 = µ, we conclude that

λ1 = |H|2 + µ

2and λ2 = |H|2 − µ

2. (4.4)

In the n = 3 case, from the Gauss equation, it follows that

K = KN (X1, X2) + |H|2 − µ2

4|H|2, (4.5)

where KN is the sectional curvature of N along M .

From (1.6) with ρ = −(logµ)/2 we get

K = −µ2

0

∆ (logµ)

= − 1

2µ

∣∣∣∣ 0

grad µ

∣∣∣∣20

− 1

2

0

∆ µ,

where ⟨·, ·⟩0 is the Euclidean metric on R2,0

∆ and0

grad are the Laplacian and the

gradient, respectively, with respect to ⟨·, ·⟩0.Therefore, replacing K in (4.5), we have that µ is a solution of (4.3).

Remark 4.27. Biconservative surfaces in BianchiCartanVranceanu spaces, which are

3-dimensional spaces with non-constant sectional curvature, were studied in [51].

Remark 4.28. If N is a 3-dimensional space form, the same result holds without the

hypothesis of biconservativity, as a CMC surface is automatically biconservative.

We note that, since K = −(∆(logµ))/2, the next result is obvious.

Corollary 4.29. Let φ :M2 → Nn be a CMC biconservative surface. Assume that M

is compact and does not have pseudoumbilical points. Then M is a topological torus.


Using Theorem 4.19 and Corollary 4.29 we obtain the following corollary.

Corollary 4.30. Let φ : M2 → Nn be a CMC biconservative surface. Assume that

M is compact and does not have pseudoumbilical points. If K ≥ 0 or K ≤ 0, then

∇AH = 0 and K = 0.

We end this section with two results which basically say that a CMC biconservative

surface in Nn can be also immersed in N3(c) having either AH or S2, as shape operator.

Theorem 4.31. Let φ :M2 → Nn be a biconservative surface. We denote by λ1 and λ2

the principal curvatures of M corresponding to φ. Assume that λ1 and λ2 are constants

and λ1 > λ2. We have:

(i) there exists locally ψ : M2 → N3(c) an isoparametric surface such that AφHφ is

the shape operator of ψ in the direction of the unit normal vector eld, where

c =µ2

4− |Hφ|4 ;

moreover,∣∣Hψ

∣∣ = |Hφ|2.

(ii) there exists locally ψ : M2 → N3(c) an isoparametric surface such that Sφ2 is the

shape operator of ψ in the direction of the unit normal vector eld, where

c = 4(µ2 − |Hφ|4

);

moreover,∣∣Hψ

∣∣ = 2 |Hφ|2.

Proof. First, we consider a symmetric tensor eld Aψ of type (1, 1) on M given by

Aψ(X) = AφHφ(X), X ∈ C(TM).

As AφHφ is a Codazzi tensor eld, Aψ satises (formally), the Codazzi equation for a

surface in a 3-dimensional space form.

Since the principal curvatures of M corresponding to φ, λ1 and λ2, are constants

and λ1 > λ2, from Proposition 4.3 it follows that K = 0. Now (formally) from the

Gauss equation for a surface in a 3-dimensional space form N3(c), and from (4.4), we

obtain

c =− detAψ

=− detAφHφ =µ2

4− |Hφ|4 .

Therefore, there exists locally an immersion ψ : M2 → N3(c) such that its shape

operator in the direction of the unit normal vector eld is Aψ. Moreover, the surface is

isoparametric as λ1 and λ2 are constants.

4.3. A Simons type formula for S2 119

We have |τ(ψ)| = 2∣∣Hψ

∣∣ = traceAψ and in the same time

traceAφHφ = λ1 + λ2 = 2 |Hφ|2 = |τ(φ)|2

2.

From the denition of Aψ one easily sees∣∣Hψ

∣∣ = |Hφ|2.Secondly, we dene the shape operator associated to the surface ψ : M2 → N3(c)

as

Aψ(X) = Sφ2 (X), X ∈ C(TM),

where c ∈ R.Since Sφ2 = −2 |Hφ|2 I + 4AφHφ , using the same argument as in the previous case,

one obtains

c =− detAψ

=− detSφ2 = 4(µ2 − |Hφ|4

)and |τ(ψ)| = |τ(φ)|2, i.e., ∣∣∣Hψ

∣∣∣ = 2 |Hφ|2 .

Theorem 4.32. Let φ : M2 → Nn be a biconservative surface. Denote by λ1 and λ2

the principal curvatures of M corresponding to φ. Assume that λ1 and λ2 are constants

and λ1 = λ2. If K = 0, then we have:

(i) there exists locally an umbilical surface ψ : M2 → N3(c) such that AφHφ is the

shape operator of ψ in the direction of the unit normal vector eld, where

c = − |Hφ|4 ;

moreover,∣∣Hψ

∣∣ = |Hφ|2.

(ii) there exists locally an umbilical surface ψ :M2 → N3(c) such that Sφ2 is the shape

operator of ψ in the direction of the unit normal vector eld, where

c = −4 |Hφ|4 ;

moreover,∣∣Hψ

∣∣ = 2 |Hφ|2.

4.3 A Simons type formula for S2

As we have already mentioned we present here some converse results of Proposition

4.1 (see Theorem 4.22, Theorem 4.39 for the compact case, and Theorem 4.41 for the

complete non-compact case).


As in the previous section, we will rst compute the rough Laplacian ∆RT for an

arbitrary symmetric tensor eld T of type (1, 1) on M with div T = 0, and then ∆RS2.


)be a surface and T a symmetric tensor eld of type

(1, 1). Assume that div T = 0. Then

trace(∇2T

)= 2KT − tKI − (∆t)I −∇ grad t. (4.6)

Proof. Let p ∈M be an arbitrary point and X1, X2 a local orthornormal frame eld,

geodesic around p. Clearly, at p we have

(trace

(∇2T

))(Xj) =

2∑i=1

(∇2T

)(Xi, Xi, Xj) .

The right hand term can be rewritten as

2∑i=1

(∇2T

)(Xi, Xi, Xj) =

2∑i=1

((∇2T

)(Xi, Xi, Xj)−

(∇2T

)(Xi, Xj , Xi)

)+

2∑i=1

(∇2T

)(Xi, Xj , Xi) .

After some straightforward computations, at p, one obtains

2∑i=1

(∇2T

)(Xi, Xi, Xj) =

2∑i=1

(Xi⟨div T,Xj⟩Xi −Xi⟨div T,Xi⟩Xj − (Xi (Xjt))Xi +

+(Xi (Xit))Xj +(∇2T

)(Xi, Xj , Xi)

),

where t = traceT .

Further, applying the Ricci formula, since div T = 0 and

2∑i=1

(∇2T

)(Xj , Xi, Xi) = 0

at p, it follows that,(trace

(∇2T

))(Xj) = (2KT − (∆t) I −∇ grad t−KtI) (Xj) .

Therefore, at p, one obtains

trace(∇2T

)= 2KT − tKI − (∆t)I −∇ grad t.

Since p was arbitrary chosen, we get that the above result holds on M .

Using (4.6), we can compute the Laplacian of the squared norm of S2 and obtain

a Simons type formula (here, instead of the second fundamental form we have the

stress-bienergy tensor).


Proposition 4.34. Let φ :M2 → Nn be a biconservative surface. Then,

12∆ |S2|2 = −2K |S2|2 + div

((⟨S2, grad

(|τ(φ)|2

)⟩)♯)

+K|τ(φ)|4

+12∆(|τ(φ)|4

)+∣∣grad (|τ(φ)|2)∣∣2 − |∇S2|2

. (4.7)

Proof. First, using the fact that traceS2 = |τ(φ)|2 and applying (4.6) with T = S2 one

obtains

∆RS2 = −2KS2 +∇ grad(|τ(φ)|2

)+(K|τ(φ)|2 +∆

(|τ(φ)|2

))I, (4.8)

where ∆RS2 = − trace(∇2S2

).

Then, from (1.2) and since M is biconservative, one gets

div(S2(grad

(|τ(φ)|2

)))= ⟨S2,Hess

(|τ(φ)|2

)⟩. (4.9)

From (1.1), with T = S = S2, it is easy to see that

1

2∆ |S2|2 =

⟨∆RS2, S2

⟩− |∇S2|2 . (4.10)

Further, since ⟨I, S2⟩ = traceS2 = |τ(φ)|2 and

∆(|τ(φ)|2

)|τ(φ)|2 = 1

2∆(|τ(φ)|4

)+∣∣grad (|τ(φ)|2)∣∣2 ,

from (4.8), (4.9), and (4.10), it follows (4.7).

Remark 4.35. If φ :M2 → Nn is a CMC biconservative surface, then S2 is a Codazzi

tensor eld and (4.8) follows from a well-known formula in [21].

Remark 4.36. Equation (4.8) was obtained, in a dierent way, in [43] but for bihar-

monic maps (a stronger hypothesis) from surfaces.

Integrating (4.7) we get the following integral formula.

Proposition 4.37. Let φ :M2 → Nn be a compact biconservative surface. Then∫M

(|∇S2|2 + 2K

(|S2|2 −

|τ(φ)|4

2

))vg =

∫M

∣∣grad (|τ(φ)|2)∣∣2 vg, (4.11)

or, equivalently,∫M

(|∇AH |2 + 2K

(|AH |2 − 2|H|4

))vg =

5

2

∫M

∣∣grad (|H|2)∣∣2 vg.

Proof. Since M is compact, by integrating (4.7) it is easy to get (4.11). To obtain the

second equation, i.e., an equivalent expression to (4.11), in terms of AH and |H|, werst recall that

|S2|2 = 16 |AH |2 − 24|H|4 (4.12)


and

∇XS2 = −2(X(|H|2

))I + 4∇XAH ,

for any X ∈ C(TM).

Then, by some standard computations, we obtain

|∇S2|2 = 16 |∇AH |2 − 24∣∣grad (|H|2

)∣∣2 .Finally, we can rewrite (4.11) as∫

M

(16 |∇AH |2 − 24

∣∣grad (|H|2)∣∣2+ (2K (16 |AH |2 − 24|H|4 − 8|H|4

))vg =

= 16

∫M

∣∣grad (|H|2)∣∣2 vg

and by a straightforward computation we get∫M

(|∇AH |2 + 2K

(|AH |2 − 2|H|4

))vg =

5

2

∫M

∣∣grad (|H|2)∣∣2 vg. (4.13)

Remark 4.38. It is easy to see that 2 |S2|2−|τ(φ)|4 = 32(|AH |2 − 2|H|4

)≥ 0, and the

equality occurs if and only if S2 =(|τ(φ)|2

/2)I, or, equivalently, M is pseudoumbilical.

From (4.13) we easily get the following result.

Theorem 4.39. Let φ : M2 → Nn be a compact CMC biconservative surface. If

K ≥ 0, then ∇AH = 0 and M is at or pseudoumbilical.

Proof. Since |H| is constant, from (4.13), one obtains

|∇AH |2 + 2K(|AH |2 − 2|H|4

)= 0.

Therefore ∇AH = 0 and K(|AH |2 − 2|H|4

)= 0. From the last equality, it follows that

K = 0, i.e., M is at, or |AH |2 − 2|H|4 = 0, i.e., M is pseudoumbilical.

Remark 4.40. Theorem 4.39 was obtained in [43] under the hypothesis of biharmon-

icity.

In the following, we study complete non-compact biconservative surfaces.

Theorem 4.41. Let φ : M2 → Nn be a complete non-compact CMC biconservative

surface with K ≥ 0. IfN

Riem ≤ k0, where k0 is a non-negative constant, then ∇AH = 0.


Proof. As |H| is constant, τ(φ) = 2H and |S2|2 = 16 |AH |2 − 24|H|4, from (4.7) we get

− 1

2∆ |S2|2 = 32K

(|AH |2 − 2|H|4

)+ |∇S2|2 . (4.14)

Since K and |AH |2 − 2|H|4 are always non-negative (see the hypothesis and Remark

4.38, respectively), we get that ∆ |S2|2 ≤ 0, i.e., |S2|2 is a subharmonic function.

Next, we show that |S2|2 is bounded from above. From (4.12) it is easy to see that

|S2|2 is bounded from above if and only if |AH |2 is bounded from above.

Let us consider X1, X2 a local orthonormal frame eld onM and η, η1, · · · , ηn−3a local orthonormal coframe eld on M such that H = |H|η. From the Gauss equation

we have

K−N

Riem (X1, X2) =⟨B (X1, X1) , B (X2, X2)⟩ − |B (X1, X2)|2

=⟨Aη (X1) , X1⟩⟨Aη (X2) , X2⟩ − (⟨Aη (X1) , X2⟩)2

+

n−3∑α=1

(⟨Aηα (X1) , X1⟩⟨Aηα (X2) , X2⟩ − (⟨Aηα (X1) , X2⟩)2

)=detAη +

n−3∑α=1

detAηα ,

whereN

Riem (X1, X2) = RN (X1, X2, X1, X2) .

It is clear that ⟨H, ηα⟩ = 0, and then traceAηα = 2⟨H, ηα⟩ = 0, for any α ∈ 1, 2, · · ·n− 3.As Aηα is symmetric, we note that detAηα ≤ 0 for any α and

∑n−3α=1 detAηα ≤ 0. Then,

we get

K−N

Riem (X1, X2) ≤ detAη. (4.15)

Let us consider µ1 and µ2 the principal curvatures of Aη. Then

detAη =µ1µ2 =(µ1 + µ2)

2 −(µ21 + µ22

)2

=(traceAη)

2 − |Aη|2

2

=4|H|2 − |Aη|2

2.

From (4.15) one obtains

|Aη|2 ≤ 4|H|2 − 2K + 2N

Riem (X1, X2) .

Since K ≥ 0 andN

Riem ≤ k0, it follows that

|Aη|2 ≤ 4|H|2 + 2k0.


Therefore, we have just proved that |AH |2 is bounded from above.

It is well known that a complete surface with K ≥ 0 is parabolic ([36]), i.e., any

subharmonic function bounded from above is constant. Thus, as |S2|2 is bounded from

above and subharmonic, it follows that |S2|2 is a constant. Using |∇S2|2 = 16 |∇AH |2

and (4.14) one obtains that ∇AH = 0, and thereforeM is either at or pseudoumbilical.

4.3.1 Exemples of submanifolds with ∇AH = 0

As we have seen, a PMC surface in a space formNn(c), n ≥ 4, is trivially biconservative.

But, if the surface is only CMC, then it is not necessarily biconservative. In [50] it

was proved that if a surface is biconservative and CMC in N4(c), with c = 0, then the

surface has to be PMC, i.e., the trivial case for our problem. We just recall here that,

if c = 1, then a PMC surface in S4 is either a minimal surface of a small hypersphere

of radius a, a ∈ (0, 1), in S4, or a CMC surface in a small or great hypersphere in S4

(see [74,75]). Of course, if we consider a CMC biconservative surface M2 of genus 0 in

R4, it is pseudoumbilical and therefore it is PMC, i.e., M2 is a 2-sphere (see [35]). In

R4, there were obtained all CMC biconservative surfaces which are not PMC. They

are given by the isometric immersion φ : R2 → R4 dened by

φ(u, v) = γ(u) + (v + a)e4,

where γ : R → R3 is a smooth curve parametrized by arc-length with positive constant

curvature κ and torsion τ = 0. By a straightforward computation we obtain that the

second fundamental form of the surface is given by

B (∂u, ∂u) = κ(u)N(u), B (∂u, ∂v) = 0, B (∂v, ∂v) = 0,

where T (u), N(u), B(u) is the Frenet frame eld of the curve γ. Then, one obtains

the expression of the mean curvature vector eld

H(u, v) =κ

2N(u),

the shape operator in the direction of H

AH (∂u) =κ2

2∂u, AH (∂v) = 0

and

∇⊥∂uH =

κ

2τ(u)N(u), ∇⊥

∂vH = 0.

It is easy to see that φ is a biconservative immersion, i.e., it satises

grad(|H|2

)+ 2 traceA∇⊥

· H(·) + 2 trace

(RR4

(·,H)·)⊤

= 0.

Therefore, φ satises all hypotheses of Theorem 4.41 which implies that AH is parallel,

a fact which can be also checked by a direct computation.

Bibliography

[1] G.B. Airy, On the strains in the interior of beams, Philos. Trans. R. Soc. London

Ser. A 153 (1863), 4979.

[2] K. Akutagawa, S. Maeta, Biharmonic properly immersed submanifolds in Euclidean

spaces, Geom. Dedicata (1) 164 (2013), 351355.

[3] A. Arvanitoyeorgos, F. Defever, G. Kaimakamis, V.J. Papantoniou, Biharmonic

Lorentz hypersurfaces in E41 , Pacic J. Math. 229 (2007), 293305.

[4] P. Baird, J. Eells, A conservation law for harmonic maps, Geometry Symposium

Utrecht 1980, 125, Lecture Notes in Math. 894, Springer, Berlin-New York, 1981.

[5] P. Baird, A. Fardoun, S. Ouakkas, Conformal and semi-conformal biharmonic

maps, Ann. Global Anal. Geom. 34 (2008), 403414.

[6] P. Baird, A. Fardoun, S. Ouakkas, Biharmonic maps from biconformal transform-

ations with respect to isoparametric functions, Dierential Geom. Appl. 50 (2017),

155166.

[7] P. Baird, D. Kamissoko, On constructing biharmonic maps and metrics, Ann.

Global Anal. Geom. 23 (2003), no. 1, 6575.

[8] P. Baird, J.C. Wood, Harmonic Morphisms Between Riemannian Manifolds, Lon-

don Mathematical Society Monographs, New Series, 29, The Clarendon Press, Ox-

ford University Press, Oxford, 2003.

[9] A. Balmu³, Biharmonic Maps and Submanifolds, DGDS Monographs 10, Geometry

Balkan Press, Bucharest, 2009.

[10] A. Balmu³, S. Montaldo, C. Oniciuc, Biharmonic PNMC submanifolds in spheres,

Ark. Mat. 51 (2013), 197221.

125

126 Bibliography

[11] A. Besse, Eistein Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete

(3) 10, Springer-Verlag Berlin, 1987.

[12] W.M. Boothby, An Introduction to Dierentiable Manifolds and Riemannian Geo-

metry, Academic Press, Inc., Orlando, Florida, 1986.

[13] R. Caddeo, S. Montaldo, C. Oniciuc, Biharmonic submanifolds of S3, Internat. J.Math. 12 (2001), 867876.

[14] R. Caddeo, S. Montaldo, C. Oniciuc, Biharmonic submanifolds in spheres, Israel

J. Math. 130 (2002), 109123.

[15] R. Caddeo, S. Montaldo, C. Oniciuc, P. Piu, Surfaces in three-dimensional space

forms with divergence-free stress-bienergy tensor, Ann. Mat. Pura Appl. (4) 193

(2014), 529550.

[16] M. do Carmo, Dierential Geometry of Curves and Surfaces, Prentice-Hall, Inc.

Englewood Clis, New Jersey, 1976.

[17] M. do Carmo, Riemannian Geometry, Birkhäuser Boston, 1992.

[18] B-Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, Series in Pure

Mathematics, 1. World Scientic Publishing Co., Singapore, 1984.

[19] B-Y. Chen, Some open problems and conjectures on submanifolds of nte type,

Soochow I. Math. 17 (1991), 169188.

[20] B-Y. Chen, S. Ishikawa, Biharmonic surfaces in pseudo-Euclidean spaces, Mem.

Fac. Sci. Kyushu Univ. Ser. A. 45 (1991), 323347.

[21] S.Y. Cheng, S.T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann.

225 (1977), 195204.

[22] M. Dajczer, Submanifolds and Isometric Immersions, Publish or Perish, Inc., 1990.

[23] J. Eells, L. Lemaire, Selected Topics in Harmonic Maps, Conf. Board Math. Sci

50, 1983.

[24] J. Eells, L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc.

20 (1988), 385524.

[25] J. Eells, A. Ratto, Harmonic Maps and Minimal Immersions with Symmetries,

Princeton University Press, 1993.

[26] J. Eells, J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J.

Math. 86 (1964), 109160.

Bibliography 127

[27] D. Fetcu, S. Nistor, C. Oniciuc, On biconservative surfaces in 3-dimensional space

forms, Comm. Anal. Geom. (5) 24 (2016), 10271045.

[28] D. Fetcu, C. Oniciuc, A.L. Pinheiro, CMC biconservative surfaces in Sn × R and

Hn × R, J. Math. Anal. Appl. 425 (2015), 588609.

[29] D. Fetcu, A.L Pinheiro, Biharmonic surfaces with parallel mean curvature in com-

plex space forms, Kyoto J. Math. (4) 55 (2015), 837855.

[30] Y. Fu, Explicit classication of biconservative surfaces in Lorentz 3-space forms,

Ann. Mat. Pura Appl. (4) 194 (2015), 805822.

[31] Y. Fu, N.C. Turgay, Complete classication of biconservative hypersurfaces with di-

agonalizable shape operator in Minkowski 4-space, Internat. J. Math. (5) 27 (2016),

1650041, 17 pp.

[32] W.B. Gordon, An analytical criterion for the completeness of Riemannian mani-

folds, Proc. Amer. Math. Soc. 37 (1973), 221225.

[33] Th. Hasanis, Th. Vlachos, Hypersurfaces in E4 with harmonic mean curvature

vector eld, Math. Nachr. 172 (1995), 145169.

[34] D. Hilbert, Die grundlagen der physik, Math. Ann. 92 (1924), 132.

[35] D. Homan, Surfaces of constant mean curvature in manifolds of constant

curvature, J. Dierential Geometry 8 (1973), 161176.

[36] A. Huber, On subharmonic functions and dierential geometry in the large, Com-

ment. Math. Helv. 32 (1957), 1372.

[37] J. Inoguchi, T. Sasahara, Biharmonic hypersurfaces in Riemannian symmetric

spaces I, Hiroshima Math. J. (1) 46 (2016), 97121.

[38] G.Y. Jiang, 2-harmonic maps and their rst and second variational formulas,

Chinese Ann. Math. Ser. A7 (4) (1986), 389402.

[39] G.Y. Jiang, The conservation law for 2-harmonic maps between Riemannian man-

ifolds, Acta Math. Sinica 30 (1987), 220225.

[40] H.B. Lawson, Complete minimal surfaces in S3, Ann. of Math. (2) 92 (1970), 335

374.

[41] R. Lopez, Constant Mean Curvature Surfaces with Boundary, Springer Monographs

in Mathematics, Berlin Heidelberg, 2013.

128 Bibliography

[42] E. Loubeau, S. Montaldo, C. Oniciuc, The stress-energy tensor for biharmonic

maps, Math. Z. 259 (2008), 503524.

[43] E. Loubeau, C. Oniciuc, Biharmonic surfaces of constant mean curvature, Pacic

J. Math. 271 (2014), 213230.

[44] Y. Luo, S. Maeta, Biharmonic hypersurfaces in a sphere, Proc. Amer. Math. Soc.

(7) 145 (2017), 31093116.

[45] M. Markellos, H. Urakawa, The biharmonicity of sections of the tangent bundle,

Monatshefte fur mathematik (3) 178 (2015), 389404.

[46] J.C. Maxwell, On reciprocal diagrams in space, and their relation to Airy's function

of stress, Proc. London. Math. Soc. 2 (1868), 102105.

[47] W.H. Meeks III, J. Perez, A Survey on Classical Minimal Surface Theory, Univer-

sity Lecture Series, vol. 60, USA, 2012.

[48] W.H. Meeks III, A. Ros, H. Rosenberg, The Global Theory of Minimal Surfaces in

Flat Spaces, Springer, Berlin Heidelberg New-York, 2000.

[49] S. Montaldo, C. Oniciuc, A. Ratto, Proper biconservative immersions in the Euc-

lidean space, Ann. Mat. Pura Appl. (4) 195 (2016), 403422.

[50] S. Montaldo, C. Oniciuc, A. Ratto, Biconservative surfaces, J. Geom. Anal. 26

(2016), 313329.

[51] S. Montaldo, I. Onnis, A.P. Passamani, Biconservative surfaces in BCV -spaces,

Math. Nachr. (2017), to appear.

[52] S. Montiel, A. Ros, Curves and Surfaces, Graduate Studies in Mathematics, vol.

69, Am. Math. Soc., 2005.

[53] A. Moroianu, S. Moroianu, Ricci surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)

14 (2015), 10931118.

[54] S. Nistor, Complete biconservative surfaces in R3 and S3, J. Geom. Phys. 110

(2016), 130153.

[55] S. Nistor, On biconservative surfaces, preprint, arXiv: 1704.04598.

[56] S. Nistor, C. Oniciuc, Global properties of biconservative surfaces in R3 and S3,Proceedings of The International Workshop on Theory of Submanifolds, Istambul

Technical University, Turkey 2-4 June 2016, to appear.

Bibliography 129

[57] S. Nistor, C. Oniciuc, On the uniqueness of complete biconservative surfaces in R3,

work in progress.

[58] S. Ohno, T. Sakai, H. Urakawa, Biharmonic homogeneous hypersurfaces in compact

symmetric spaces, Dierential Geom. Appl. 43 (2015), 155179.

[59] C. Oniciuc, Biharmonic maps between Riemannian manifolds, An. Stiint. Univ.

Al.I. Cuza Iasi Mat (N.S.) 48 (2002), 237248.

[60] C. Oniciuc, O Introducere în Teoria Aplicaµiilor Armonice, Casa Editorial Demi-

urg, Ia³i, 2007.

[61] C. Oniciuc, Biharmonic Submanifolds in Space Forms, Habilitation Thesis,

http://www.math.uaic.ro/~oniciucc/TE-2011-3-0108_web/

Oniciuc_TezaAbil-Web.pdf, 2012.

[62] Y.L. Ou, Biharmonic hypersurfaces in Riemannian manifolds, Pacic J. Math. 248

(2010), 217232.

[63] P. Petersen, Riemannian Geometry, Springer, USA, 2006.

[64] G. Ricci-Curbastro, Sulla teoria intrinseca delle supercie ed in ispecie di quelle di

2 grado, Ven. Ist. Atti (7) VI (1895), 445488.

[65] M. Sakaki, Minimal surfaces with the Ricci condition in 4-dimensional space forms,

Proc. Amer. Math. Soc. 121 (1994), 573577.

[66] A. Sanini, Applicazioni tra varietà riemanniane con energia critica rispetto a de-

formazioni di metriche, Rend. Mat. 3 (1983), 5363.

[67] T. Sasahara, Surfaces in Euclidean 3-space whose normal bundles are tangentially

biharmonic, Arch. Math. (Basel) (3) 99 (2012), 281287.

[68] T. Sasahara, Tangentially biharmonic Lagrangian H-umbilical submanifolds in com-

plex space forms, Abh. Math. Semin. Univ. Hambg. 85 (2015), 107123.

[69] N.C. Turgay, H-hypersurfaces with three distinct prinicipal curvatures in the Euc-

lidean spaces, Ann. Math. 194 (2015), 17951807.

[70] A. Upadhyay, N.C. Turgay, A Classication of Biconservative Hypersurfaces in a

Pseudo-Euclidean Space, J. Math. Anal. Appl. (2) 444 (2016), 17031720.

[71] H. Urakawa, Calculus of Variations and Harmonic Maps, Translations of Mathem-

atical Monographs, vol. 132, 1993.

http://www.math.uaic.ro/~oniciucc/TE-2011-3-0108_web/Oniciuc_TezaAbil-Web.pdf

http://www.math.uaic.ro/~oniciucc/TE-2011-3-0108_web/Oniciuc_TezaAbil-Web.pdf

130 Bibliography

[72] H. Urakawa, Biharmonic maps into compact Lie groups and integrable systems,

Hokkaido Math. J. (1) 43 (2014), 73103.

[73] Th. Vlachos, Minimal surfaces, Hopf dierentials and the Ricci condition,

Manuscripta Math. (2) 126 (2008), 201230.

[74] S.T. Yau, Submanifolds with constant mean curvature I, Amer. J. Math. 96 (1974),

346366.

[75] S.T. Yau, Submanifolds with constant mean curvature II, Amer. J. Math. 97 (1975),

76100.

Alexandru Ioan Cuza University of Ia³isbarna/resurse/Nistor_TezaDoctorat.pdf · Alexandru Ioan...

Documents

Transcript of Alexandru Ioan Cuza University of Ia³isbarna/resurse/Nistor_TezaDoctorat.pdf · Alexandru Ioan...