I) Introduction1 to CR geometry

and subelliptic harmonic maps.

II) Boundary values

of Bergman-harmonic maps

Sorin Dragomir2

Abstract. We give an elementary introduction to CR and pseu-dohermitian geometry, starting from H. Lewy’s legacy (cf. [20])i.e. tangential Cauchy-Riemann equations on the boundary of theSiegel domain. In this context we describe fundamental objects,such as contact structures, Levi forms, the Tanaka-Webster con-nection and the Fefferman metric (cf. e.g. [4]). Also naturallyarising Hormander systems of vector fields, the associated sub-laplacians, and J. Jost and C-J. Xu’s subelliptic harmonic maps(cf. [16]), a first geometric interpretation of which is given in termsof Lorentzian geometry (cf. [2]). A second, more specialized talk -scheduled for the afternoon of the same day - is devoted to the dis-cussion of boundary values of Bergman-harmonic maps. There westart from A. Koranyi & H.M. Reimann’s crucial observation (cf.[17]) that, as a consequence of Fefferman’s asymptotic expansionformula (cf. [9]) for the Bergman kernel of a smoothly boundedstrictly pseudoconvex domain Ω ⊂ Cn

K(ζ, z) = CΩ|∇ϕ(z)|2 · detLϕ(z) ·Ψ(ζ, z)−(n+1) + E(ζ, z),

|E(ζ, z)| ≤ C ′Ω |Ψ(ζ, z)|−(n+1)+1/2 · |log |Ψ(ζ, z)|| ,the Kahlerian geometry of the interior of Ω may be effectivelyrelated to the contact geometry of its boundary ∂Ω. Then wemake use of the Graham-Lee connection (cf.[13]) to derive the com-patibility equations on ∂Ω satisfied by the boundary values of aBergman-harmonic map Φ : Ω→ S which is C∞ up the boundary.We are led to a geometric interpretation (cf. [5]) of Jost & Xu’ssubelliptic harmonic maps from an open set U ⊂ R2n−1 carryinga given Hormander system of vector fields.

1Talks given at The State University of New Jersey Rutgers Campus at Camden,Department of Mathematical Sciences, within A Day for Complex Analysis andGeometry, May 3, 2013.

2Dipartimento di Matematica, Informatica ed Economia, Universita degli Studidella Basilicata, Viale dell’Ateneo Lucano 10, Campus Macchia Romana, 85100Potenza, Italy, e-mail sorin.dragomir@unibas.it

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2

1. Tangential Cauchy-Riemann equations: Hans Lewy’slegacy

Let us start with the Siegel domain

Ωn = (z, w) ∈ Cn × C : Im(w) > |z|2and consider the Dirichlet problem for the ordinary Cauchy-Riemannsystem

(1) ∂F = 0 in Ωn,

(2) F = f on ∂Ωn,

for some f ∈ C∞(∂Ωn,C). Let us look at the C∞ regularity up tothe boundary of the solution to (1)-(2) i.e. assume that a solutionF ∈ C∞(Ωn,C) exists. If

ρ(z, w) =1

2i(w − w)−

n∑j=1

zjzj

and ε > 0 then Mε = (z, w) ∈ Cn+1 : ρ(z, w) = ε is a smooth realhypersurface lying in the interior of Ωn. A complex tangent vector field

Z =n∑j=1

λj∂

∂zj+ µ

∂

∂w

is tangent to Mε if and only if Z(ρ) = 0 i.e. µ = 2i∑

j λjzj. Complex

vector fields, of type (1, 0), which are tangent to Mε are therefore ofthe form Z =

∑j λjLj where

Lj ≡∂

∂zj+ 2izj

∂

∂w.

As F is holomorphic in Ωn

∂F

∂zj= 0,

∂F

∂w= 0,

one has Lj(F ) = 0 and therefore along Mε. Let us tend with ε → 0,which is the same to approach the boundary as (z, w) → ∂Ωn. Itfollows that the boundary values f of F must satisfy the equations

(3) Ljf = 0.

In the language of PDEs, C∞ regularity up to the boundary in theDirichlet problem (1)-(2) doesn’t come for free, the boundary data mustsatisfy certain compatibility equations along ∂Ωn i.e. equations (3).These are the tangential Cauchy-Riemann equations (on the bound-ary of the Siegel domain) and Lj are the Lewy operators. Indeed the

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operator L1 has been discovered by H. Lewy in his fundamental pa-per of 1958 (while investigating the boundary behavior of holomorphicfunctions on the Siegel domain in C2, cf. [20]).

The span of Lj(p) : 1 ≤ j ≤ n at each p ∈ ∂Ωn gives a com-plex rank n subbundle T1,0(∂Ωn) of the complexified tangent bundleT (∂Ωn)⊗ C, the CR structure of ∂Ωn satisfying

(4) T1,0(M)p ∩ T0,1(M)p = (0), p ∈M ≡ ∂Ωn ,

(5) Z,W ∈ C∞(U, T1,0(M)) =⇒ [Z,W ] ∈ C∞(U, T1,0(M)),

where T0,1(∂Ωn) = T1,0(∂Ωn) and U ⊂ ∂Ωn is an open set. Accordingto the calculations above, the CR structure T1,0(∂Ωn) is induced by thecomplex structure of the ambient space Cn+1 in the sense that

T1,0(∂Ωn)p = [Tp(∂Ωn)⊗R C] ∩ T 1,0(Cn+1

)p, p ∈ ∂Ωn,

where T 1,0(Cn+1) is the holomorphic tangent bundle over Cn+1. Clearlythese considerations admit a verbatim repetition to make sense of anotion of induced CR structure on the boundary of any domain Ω ⊂Cn+1 (whose boundary is smooth), and actually on any smooth realhypersurface M ⊂ Cn+1.

Let M ⊂ Cn+1 be a real hypersurface, endowed with the inducedCR structure T1,0(M), and let us consider the first order differentialoperator

∂b : C∞(M,C)→ C∞ (T0,1(M)∗) ,

(∂bf)L = L(f), f ∈ C∞(M,C), L ∈ T1,0(M).

This is the tangential Cauchy-Riemann operator and ∂bf = 0 are thetangential Cauchy-Rimann equations on M (clearly equivalent to (3)when M is the boundary of the Siegel domain). A solution to ∂bf = 0is a Cauchy-Riemann function. We shall not worry at this time for theregularity to be requested a priori, or to be expected a posteriori, forsuch a CR function. Note that for any holomorphic function F ∈ O(U),defined on an open set U ⊂ Cn+1 such that U ∩ M 6= ∅, the tracef = F |U∩M is a CR function on U ∩M . We assume tacitly that Mis embedded, so that U ∩M be open in M . The converse is one ofthe main problems in CR geometry (the CR extension problem) andclearly admits a local, as well as a global, formulation.

Let (M,T1,0(M)) be an abstract CR manifold i.e. a real (2n + 1)-dimensional C∞ manifold endowed with the a complex rank n sub-bundle T1,0(M) ⊂ T (M)⊗C obeying to (4)-(5). Another fundamentalproblem in CR geometry (the embedding problem) is whether an embed-ding of M in Cn+1 exists such that T1,0(M) be the CR structure induced

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by the complex structure of Cn+1. The embedding problem clearly ad-mits a local version as well and is still related to the tangential Cauchy-Riemann equations, for given an embedding f = (f 1, · · · , fn+1) : M →Cn+1 such that

(dpf)T1,0(M)p = [Tp(M)⊗R] ∩ T(Cn+1

)f(p)

, p ∈M,

it may be easily seen that each f j is a CR function on M . Hence, tosolve the embedding problem one needs to produce enough functionallyindependent solutions to ∂bf = 0.

Let us go back to the boundary of the Siegel domain Ω1 ⊂ C2.Precisely, let H = C×R be the Heisenberg group i.e. the 3-dimensionalLie group with the law

(z, t) · (w, s) = (z + w, t+ s+ 2 Im(z w)), (z, t), (w, s) ∈ H.

The map

f : H→ ∂Ω1, f(z, t) = (z, t+ i |z|2), (z, t) ∈ H,

is a C∞ diffeomorphism of H onto the boundary of the Siegel domainand f−1 maps the Lewy operator L1 into L where

L =∂

∂z+ i z

∂

∂t.

The span of L over C is a CR structure T1,0(H) and then f is a CRisomorphism i.e. a diffeomorphism such that

(dpf)T1,0(H) = T1,0(∂Ω1)f(p) , p ∈ H.

Lewy’s operator L possesses well known unsolvability properties i.e.there is a smooth f such that Lu = f has no smooth solution inany neighborhood of the origin, eliminating a belief (popular at thebeginning of the 1950s) that linear first order equations with smoothcoefficients should always have solutions (of course nonlinear examplestrivially exist e.g. eu

′= 0 has no solutions). Solvability turns to be

closely related to (local) embeddability as shown by Hill’s example(cf. [14]). The equation Lu = ω (with ω ∈ C∞(H,C) is solvable at(z0, t0) ∈ R3 if there is an open set U ⊂ R3 and a function u ∈ C∞(U,C)such that Lu = ω in U . Let T1,0(H× C) be the CR structure spannedby

P =∂

∂ζ, Q = L+ ω(z, t)

∂

∂ζ,

where ζ : H×C→ C is the projection. The result of C.D. Hill alludedto above is that T1,0(H× C) is locally embeddable at (z0, t0, ζ0) if andonly if Lu = ω is solvable at (z0, t0).

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Let Ψ : M → Cn be a C1-embedding of a real m-dimensional mani-fold M . A point x ∈M is a complex tangent of Ψ(M) if

[(dxΨ)Tx(M)] ∩[JΨ(x)(dxΨ)Tx(M)

]6= (0)

where J is the complex structure on Cn. Let ℵΨ(M) be the set of allcomplex tangents of Ψ(M). For each x ∈ ℵΨ(M) let H(M)x be themaximal complex subspace of Tx(M) i.e.

(dxΨ)H(M)x = [(dxΨ)Tx(M)] ∩[JΨ(x)(dxΨ)Tx(M)

].

The dimension dimCH(M)x is the degree of x. When Ψ(M) is a realhypersurface in Cn one has ℵΨ(M) = M and each complex tangentx ∈ M has degree n − 1. In general ℵΨ(M) has a rather complicatedtopological structure (e.g. it is singular, stratified, etc.).

An embedding Ψ : M → Cn is totally real if ℵΨ(M) = ∅ and ifthis is the case it must be that m ≤ n, and the case m = n is ofparticular interest. Not all real n-dimensional manifolds admit totallyreal embeddings in Cn. For instance, by a result of M. Gromov theonly spheres Sn admitting a totally real embedding Sn → Cn are S1

and S3. Ahern and Rudin exhibited an explicit totally real embeddingS3 → C3 (cf. [1]).

A knot is a continuous simple embedding of S1. Two knots Ki ⊂ S3

(i = 1, 2) are topologically equivalent, or have the same topological type,if there is a homeomorphism h : S3 → S3 such that h(K1) = K2. Abeautiful result by A.L. Elgindi (cf. [8]) is that for every knot K ⊂ S3

and every positive integer n ∈ N there is an embedding Ψ : S3 → C3 ofclass Cn such that ℵΨ(S3) is a knot topologically equivalent to K. Thatis every topological type of a knot in S3 arises as the set of all complextangents to some embedding of S3 into C3. Remarkably the proof ofElgindi’s result relies on the following facts

i) If M = ρ−1(0) ⊂ C2 is a real hypersurface and Ψ : M → C3

the embedding of M in C3 as the graph of a given smooth functionf : M → C

Ψ(z, w) = (z, w, f(z, w)), (z, w) ∈M,

then

ℵΨ(M) =(Lf)−1

(0).

ii) The map L : C [z, w, z, w]→ C [z, w, z, w] is (linear and) surjective.A moral conclusion is that a deep circle of ideas relates the notions

above and that a research project titled say Complex tangents, knotsand analytic discs, solvability and embeddability, through the geometryof canonical bundles would be as actual today as well as at the time ofH. Jacobowitz’s results on this matter (cf. [15]).

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2. Hormander systems and the Jost-Xu program

Let

L =1

2(X − iY ) ,

X =∂

∂x+ 2y

∂

∂t, Y =

∂

∂y− 2x

∂

∂t,

be the real and imaginary parts of L. The vector fields X, Y, ∂/∂tare left invariant and form a basis of the Lie algebra of H. Note that

[X, Y ] = −4∂

∂t

(Heisenberg’s commutation relation) hence the vector fields X, Y andtheir commutator span the tangent space to H = R3 at each point.Hence X, Y is a special instance of what one commonly calls aHormander system on R3. Let X∗ , Y ∗ be the formal adjoints ofX, Y e.g. ∫

HX∗f ϕ dµ =

∫Hf Xϕ dµ

for all f, ϕ ∈ C10(H) (actually X∗ = −X and Y ∗ = −Y yet irrelevant).

Next let us consider the second order differential operator

∆bu = X∗Xu+ Y ∗Y u

(commonly referred to as a Hormander sum of squares). Let Ω ⊂ H bea bounded domain with smooth boundary. The space DO2(Ω) of allsecond order differential operators

A =∑|α|≤2

Aα(x)Dα

with real valued continuous coefficients on Ω, is a Banach space withthe norm

‖A‖ =∑|α|≤2

supx∈Ω

|Aα(x)| .

If EO2(Ω) is the subset consisting of all second order differential oper-ators which are elliptic in Ω then ∆b ∈ ∂ EO2(Ω) i.e. ∆b is a boundarypoint of EO2(Ω) and ∆b is a degenerate elliptic operator in this sense.More explicitly, a calculation of the symbol of ∆b shows that ellipticitydegenerates precisely in the cotangent directions θ0,p, p ∈ H, where

(6) θ0 = dt+ in∑j=1

zj dzj − zj dzj

.

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However ∆b is subelliptic of order ε = 1/2 i.e. for every point p ∈ Hthere is an open neighborhood U ⊂ H of p such that

‖u‖21/2 ≤ C

(|(∆bu , u)L2|+ ‖u‖2

L2

), u ∈ C∞0 (U),

where ‖ · ‖ε is the Sobolev norm of order ε

‖u‖ε =

(∫R3

(1 + |ξ|2

)ε |u(ξ)|2 dξ) 1

2

(u is the Fourier transform of u). Then, by a result of J.J. Kohn, ∆b

is hypoelliptic i.e. if ∆bu = f in distributional sense and f ∈ C∞ thenu ∈ C∞ as well. Hypoellipticity is the main property that ∆b shareswith elliptic operators.

Let N be a Riemannian manifold, carrying the Riemannian metrich, which may be covered by a single coordinate neighborhood χ =(y1, · · · , ym) : N → Rm, and let us consider the energy functional

EΩ : C∞(H, N)→ R,

EΩ(φ) =1

2

∫Ω

2∑a=1

Xa(φj)Xa(φ

k) (hjk φ) dµ ,

where X1 = X and X1 = Y and µ is the Lebesgue measure on R3.Also

φj = yj φ, hjk = h(∂/∂yj , ∂/∂yk), 1 ≤ j, k ≤ m.

The Euler-Lagrange equations of the variational principle δ EΩ(φ) = 0are

(7) (HNφ)i ≡ −∆bφi +

2∑a=1

Xa(φj)Xa(φ

k)(Γijk φ

)= 0

and a solution φ ∈ C∞(H, N) to (7) is a subelliptic harmonic map. Thenotion is due to J. Jost & C-J. Xu, who started a program of recoveringresults known for quasilinear elliptic systems of variational origin, suchas the harmonic map system, to the at least hypoelliptic case. Clearly(7) can be made sense of for an arbitrary Hormader system of vectorfields, on an open subset U ⊂ RN , and if these vector fields are thecoordinate vector fields ∂/∂xA : 1 ≤ A ≤ N a subelliptic harmonicmap is nothing but a harmonic map φ : U → N . Also a notion ofsubelliptic harmonic morphism is ready available for one may considercontinuous maps φ : H → N pulling back local harmonics on N intolocal harmonics of ∆b i.e.

∀ v : V ⊂ N → R, ∆hv = 0 in V =⇒ ∆b(v φ) = 0 in φ−1(V ).

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Here ∆h is the ordinary Laplace-Beltrami operator of (N, h) on func-tions. Just as in the classical case of harmonic morphisms among Rie-mannian manifolds φ : H → N must be smooth. Indeed one maychoose, about each point of N , a harmonic local coordinate system(V, yj) and then φj = yj φ must be harmonics of ∆b i.e. ∆bφ

i = 0.Yet ∆b is hypoelliptic hence each φi, and then φ, is C∞.

So Jost & Xu’s subeliptic harmonic maps are no doubt a generaliza-tion of harmonic maps, and just as S. Campanato, M. Giaquinta, S.Hildebrand, H. Kaul, and K. Widman at their time, Jost & Xu maysolve the Dirichlet problem for maps with values in regular balls, ex-hibiting a wonderful use of the so called subelliptic technicalities bornwith the work by E. Lanconelli and considerably developed into anautonomous and respectable chapter of PDEs theory by a number ofpeople, among which the most important is perhaps N. Garofalo. Jost& Xu’s result (cf. [16]) may be loosely stated as follows.

Theorem 1. Let Ω ⊂ H be a bounded domain with smooth boundary.Let (N, h) be a complete Riemannian manifold, covered by a single co-ordinate neighborhood. Let us assume the sectional curvature of (N, h)is bounded by above by a constant κ2 and let p ∈ N be a point and0 < µ < minπ/(2κ) , i(p) where i(p) is the injectivity radius of p.Let f ∈ C

(Ω, N

)∩W 1,2

X,Y (Ω, N) such that

f(Ω)⊂ B(p, µ) = q ∈ N : dh(p, q) < µ.

Then there is a unique map φ ∈ W 1,2X,Y (Ω, N) ∩ L∞(Ω, N) such that

φ|∂Ω = f and φ(Ω) ⊂ B(p, µ) and φ minimizes EΩ among such maps.Also this φ is a weak solution to HNφ = 0. Moreover φ has the sameinterior regularity properties as solutions to linear hypoelliptic systemsand if ∂Ω is C∞ and noncharacteristic for X, Y and if f is C∞ onegets the corresponding boundary regularity of φ.

Here W 1,2X,Y (Ω, N) are Sobolev-type spaces of maps whose compo-

nents admit weak L2 derivatives in the directions X and Y . Theboundary ∂Ω is noncharacteristic for the system X, Y if for everypoint p ∈ ∂Ω either Xp 6∈ Tp(∂Ω) or Yp 6∈ Tp(∂Ω). It may be easilyshown (e.g. by looking at generalized solutions to ∆bu = f i.e. so-lutions u ∈ C1 whose second order derivatives have jumps across ∂Ω)that the (non) characteristic notion just introduced is the ordinary no-tion in the theory of characteristics and of the Cauchy problem for∆bu = f .

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Our point of view in this talk is that Jost & Xu’s notion (of a subel-liptic harmonic map) is rather formal1 (in the end just replacing by∆b the ordinary Laplacian in the harmonic map system) and wish toexhibit a geometric interpretation.

The differential 1-form θ0 given by (6) above is a contact form onH i.e. θ0 ∧ dθ0 is a volume form on H. It is but one of the infinitelymany nowhere zero sections of the conormal bundle H(H)⊥ ⊂ T ∗(H)where H(H) is the Levi distribution of H as a CR manifold, with theCR structure T1,0(H). Precisely

H(H) = Re T1,0(H) ∩ T0,1(H) ,H(H)⊥p = ω ∈ T ∗p (H) : Ker(ω) ⊃ H(H)p, p ∈ H.

Then H(H)⊥ → H is a real line bundle (as H is oriented and connectedthis bundle is actually trivial i.e. H(H)⊥ ≈ H × R) and any othernowhere zero section θ ∈ C∞(H(H)⊥) (any pseudohermitian struc-ture on H, according to the terminology introduced by S. Webster)is ”conformally” related to θ0 i.e. θ = λ θ0 for some C∞ functionλ : H→ R \ 0. In particular any pseudohermitian structure on H isa contact form. The Levi distribution carries the complex structure

J : H(H)→ H(H),

J(Z + Z) = i(Z − Z), Z ∈ T1,0(H).

Given any contact form θ ∈ C∞(H(H)⊥) the Levi form is

Gθ(X, Y ) = (dθ)(X, JY ), X, Y ∈ H(H).

As an immediate consequence of definitions

Gθ = λGθ0

accounting for the (largely exploited) analogy among CR and conformalgeometry. An easy verification shows that Gθ0 is positive definite hence(H, T1,0(H)) is strictly pseudoconvex, a notion which is immediately

1Although the functional EΩ : C∞(H, N)→ R above isn’t postulated but ratherdiscovered by setting

Xj =

3∑A=1

bAj (x)∂

∂xA, aAB(x) =

2∑j=1

bAj (x)bBj (x),

(gε)AB

= aAB + ε δAB , [(gε)AB ] =[(gε)

AB]−1

, ε > 0,

EΩ,ε(φ) =1

2

∫Ω

‖dφ‖2gε dx =1

2

∫Ω

(gε)AB ∂φα

∂xA∂φβ

∂xB(hαβ φ) dx

(the ordinary Dirichlet functional on smooth maps among the Riemannian man-ifolds (M, gε) with gε(∂A , ∂B) = (gε)AB and (N,h)) and letting ε → 0 yieldsEΩ,ε(φ)→ EΩ(φ).

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made sense of on an abstract CR manifold. If θ is chosen such that Gθ

be positive definite (i.e. λ > 0) then one may consider the Riemannianmetric gθ (the Webster metric) given by

gθ(X, Y ) = Gθ(X, Y ), gθ(X,T ) = 0, gθ(T, T ) = 1,

for any X, Y ∈ H(H). Here T ∈ X(H) is the Reeb vector of (H, θ) i.e.the tangent vector field on H determined by

θ(T ) = 1, T c dθ = 0.

Then (H(H), Gθ) is a sub-Riemannian structure on H and the Webstermetric gθ contracts the sub-Riemannian structure i.e.

dgθ(p, q) ≤ dH(p, q), p, q ∈ H,

where dgθ and dH are respectively the distance function associated tothe Riemannian structure gθ and the Carnot-Cartheodory distancefunction. A rich geometric structure arises from the CR structureT1,0(H) alone. To this we shall add in a moment geometric objectssuch as the Tanaka-Webster connection, and the Fefferman metric. Itis an accepted philosophy that the study of these geometric objectsmay ultimately shed light on the properties of the solutions to the tan-gential Cauchy-Riemann equations. We shall use these objects, andothers to come, such as the Graham-Lee connection, in order to givethe geometric interpretation of subelliptic harmonic maps alluded toabove.

It should be mentioned that, in a couple of papers published atthe end of the 1970’s in Trans. A.M.S and Proc. A.M.S. (cf. [3])A Bejancu also introduced a notion of CR submanifold M of a givenKahlerian manifold M , as a real submanifold carrying a distributionD such that i) JD ⊂ D and ii) the orthogonal complement D⊥ of D in(T (M), g) satisfies JD⊥ ⊂ T (M)⊥. Here J and g are respectively thecomplex structure and the Kahlerian metric on M . The notion wassoon embraced by the portion of mathematical community devotedto the study of the geometry of second fundamental form of isometricimmersions among Riemannian manifolds, a large amount of papers onCR submanifolds of Kahlerian manifolds first and then of Hermitianmanifolds of sorts (such as locally conformal Kahler manifolds) havebeen published, and eventually A. Bejancu got the Romanian Academyprice for his discovery. An observation by D.E. Blair and B-Y. Chenshows that, as a consequence of the Kahler condition on g

X − iJX ∈ T (M)⊗ C : X ∈ D

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is a CR structure on M , though of higher CR codimension (actuallya locally conformal Kahler metric g suffices). So it seems a circle hasbeen closed, and A. Bejancu’s notion does fit into CR geometry, in spiteof some criticism formulated at the time of its discovery (both becauseBlair and Chen’s observation appeared somewhat later, and because A.Bejancu’s CR submanifolds were already embedded, while it appears hewas unaware of when CR manifolds embed e.g. unaware of Andreottiand Hill’s embedability theorem). It should be however observed thatthe study of the geometry of the second fundamental form of a CRsubmanifold of a Hermitian manifold, based as it is on the use of thefirst fundamental form and of Gauss-Weingarten and Gauss-Codazzi-Ricci equations, is confined to Riemannian geometry: if for instanceM = Cn and M is the boundary of the Siegel domain then none ofthe infinitely many Webster metrics of M coincides with the first fun-damental form (the metric induced on M by the flat Kahler metric ofthe ambient space Cn). In turn the Webster metrics gθ : θ ∈ P of a(at least nondegenerate) CR manifold and their curvature describe thepseudoconvexity properties of M , as understood in complex analysisin several complex variables.

3. Fefferman’s metric

Let (M,T1,0(M)) be a CR manifold, of real dimension 2n+1 and CRdimension n. Let H(M) be its Levi distribution. Assume that M isstrictly pseudoconvex i.e. the Levi form Gθ is positive definite for somepseudohermitian structure θ on M . Let T ∈ X(M) be the Reeb vectorof (M, θ) and gθ the Webster metric. By a result got independently byS. Webster and N. Tanaka, there is a unique linear connection ∇ on Msuch that i) H(M) is parallel with respect to ∇, ii) ∇J = 0, ∇gθ = 0,and iii) the torsion tensor field T∇ of ∇ is pure i.e.

T∇(Z,W ) = 0, T∇(Z,W ) = 2iGθ(Z,W ), Z,W ∈ T1,0(M),

τ J + J τ = 0, τ(X) ≡ T∇(T,X), X ∈ X(M).

∇ is the Tanaka-Webster connection of (M, θ) (cf. [21], [22]). Theassumption of nondegeneracy of some Levi form Gθ (and thus of all)actually suffices to prove existence and uniqueness of such ∇ (we shallonly exploit the result in the strictly pseudoconvex cases).

A complex valued differential p-form ω on M is a (p, 0)-form ifT0,1(M) cω = 0. Let Λp,0(M) → M be the relevant bundle. ThenK(M) = Λn+1,0(M) is a complex line bundle over M (the canonicalline bundle). Let us set

C(M) = [K(M) \ zero section] /R+ , R+ = GL+(1,R),

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so that to obtain a principal S1-bundle

S1 → C(M)π−→M

(the canonical circle bundle). Let Fθ ∈ Lor(C(M)) be the Lorentzianmetric (the Fefferman metric of (M, θ)) given by

Fθ = π∗Gθ + 2 (π∗θ) σ,

σ =1

n+ 2

dγ + π∗

[i ωα

α − i gαβ dgαβ +ρ

4(n+ 1)θ

],

Gθ(X, Y ) = Gθ(X, Y ), Gθ(T, · ) = 0, X, Y ∈ H(M),

∇Tβ = ωβαTα , ρ = gαβRαβ ,

gαβ = Gθ(Tα, Tβ),[gαβ]

=[gαβ]−1

,

Rαβ = Ric∇(Tα, Tβ), Ric∇(X, Y ) = traceZ 7−→ R∇(Z, Y )X

,

for any local frame Tα : 1 ≤ α ≤ n ⊂ C∞(U, T1,0(M)). Also γ is alocal fibre coordinate on C(M). By a result of C.R. Graham, σ is a

connection 1-form in the principal bundle S1 → C(M)π→ M . By a

result of J.M. Lee

θ = euθ =⇒ Fθ = euπFθ

and then euπFθ : u ∈ C∞(M) (the restricted conformal class of Fθ)is a CR invariant. None of the Fefferman metrics Fθ is Einstein, yet(C(M), Fθ) is a space-time with the time orientation T ↑−S (here S isthe tangent to the action of S1 on C(M) and T ↑ is the horizontal liftof T with respect to the connection 1-form σ).

Let us assume for simplicity that M is compact (hence C(M) iscompact) and consider the energy functional

E(Φ) =1

2

∫C(M)

traceFθ (Φ∗h) dvol(Fθ),

Φ ∈ C∞(C(M), N),

where (N, h) is a Riemannian manifold. If Fθ were a Riemannian met-ric, the trace traceFθ (Φ∗h) would be the (squared) Hilbert-Schmidtnorm ‖dΦ‖2. While harmonic maps theory arises in the Riemann-ian category, it is long consolidated within the semi-Riemannian cat-egory, as well (starting with the work by B. Fuglede). A C∞ mapΦ : C(M)→ N is harmonic if

d

dtE(Φt)t=0 = 0

13

for any smooth 1-parameter variation Φt|t|<ε ⊂ C∞(C(M), N) of Φi.e. Φ0 = Φ. The Euler-Lagrange equations of δ E(Φ) = 0 are

(8) −Φi +(Γijk Φ

) ∂Φj

∂uA∂Φk

∂uB(Fθ)

AB = 0

where is the Laplace-Beltrami operator of (C(M), Fθ) (the waveoperator) and u0 = γ, uA = xAπ, 1 ≤ A ≤ 2n+1, are local coordinateson C(M) (induced by the local coordinate system (U, x1, · · · , x2n+1)on M). If Φ : C(M) → N is S1-invariant and φ : M → N is thecorresponding base map (so that Φ = φ π) then one may integrateover the fibres in E(φ π) so that to get2

E(φ π) = π

∫M

traceGθ ΠH (φ∗h) θ ∧ (dθ)n ≡ 2π E(φ)

where ΠH (φ∗h) is the restriction of φ∗h to H(M) ⊗ H(M). One hasthen discovered the energy integral which is appropriate for subelliptictheory, for φ ∈ C∞(M,N) is a critical point of E if and only if φ π isa harmonic map of (C(M), Fθ) into (N, h) (by a result of E. Barlettaet al.) and the Euler-Lagrange equations of δ E(φ) = 0 are

(9) −∆bφi +

2n∑a=1

Xa(φj)Xa(φ

k)(Γijk φ

)= 0

where Xa : 1 ≤ a ≤ 2n ⊂ C∞(U,H(M)) is a local Gθ-orthonormalframe. Equations (9) are actually the projection on M via π of theharmonic map system (8) since (by a result of J.M. Lee) π∗ = ∆b.Here ∆b (the sublaplacian of (M, θ)) is the second order differentialoperator given by

∆bu = −div(∇Hu

), u ∈ C2(M),

LX (θ ∧ (dθ)n) = div(X) θ ∧ (dθ)n,

∇Hu = ΠH∇u , ΠH : T (M) = H(M)⊕ RT → H(M),

gθ(∇u , X) = X(u), X ∈ X(M).

On the other hand, if U is also the domain of a local chart (U,ϕ) on M ,then it may be easily seen that ϕ∗Xa : 1 ≤ a ≤ 2n is a Hormandersystem on ϕ(U) ⊂ R2n+1 and ∆b ≡

∑2na=1 X

∗aXa, hence solutions to (9)

are subelliptic harmonic maps (giving a first geometric interpretationto those).

It should be mentioned that Fefferman’s metric was originally builtin the following extrinsic manner. Let Ω ⊂ Cn be a smoothly bounded

2The second π is the irrational number π ∈ R \ Z (so does the third).

14

strictly pseudoconvex domain and let us consider the Dirichlet problemfor the complex Monge-Ampere equation

(10) J(u) ≡ (−1)n det

(u ∂u/∂zk

∂u/∂zj ∂2u/∂zj ∂zk

)= 1 in Ω,

(11) u = 0 on ∂Ω.

By a result of S-Y. Cheng & S-T. Yau the problem (10)-(11) admitsa unique solution u which is C∞ in the interior of Ω and belongs toCn+(3/2)−ε(Ω). Let us consider the function

H : Ω× (C \ 0)→ R,

H(z, ζ) = |ζ|2/(n+1) u(z), z ∈ Ω, ζ ∈ C \ 0,where u is the unique solution to (10)-(11). Moreover we consider the(0, 2)-tensor field G on Ω× (C \ 0) given by

G =n∑

A,B=0

∂2H

∂zA ∂zBdzA dzB

where z0 = ζ. Then G is a biholomorphic invariant of Ω, in the fol-lowing sense. Let F : Ω → Ω be a biholomorphic map and let usset

F : Ω× (C \ 0)→ Ω× (C \ 0) ,

F(z, ζ) =

(F (z) ,

ζ

det F ′(z)

), z ∈ Ω, ζ ∈ C \ 0.

Then (by a result of C. Fefferman) F is a biholomorphism of Ω ×(C \ 0) in itself and F∗G = G. One may also show that

det

(∂2H

∂zA ∂zB

)=

J(u)

(n+ 1)2

hence G is nondegenerate and actually a semi-Riemannian metric onΩ× (C \ 0). The explicit expression of G is

G =u(z)

(n+ 1)2|ζ|2/(n+1)−2 dζ dζ+

+|ζ|2/(n+1)

n+ 1(∂u) (

1

ζdζ) +

|ζ|2/(n+1)

n+ 1(1

ζdζ) (∂u)+

+|ζ|2/(n+1) ∂2u

∂zj ∂zkdzj dzk .

Let Ω be given by the defining function ϕ i.e. Ω = ϕ < 0 andDϕ(z) 6= 0 for every z ∈ ∂Ω. For each ε ≥ 0 let us set Mε = z ∈ Ω :ϕ(z) = −ε (so that M0 = ∂Ω). Let Gε = j∗εG where jε : Mε × S1 →

15

Ω × (C \ 0) is the inclusion. It may be shown that Gε tends, asε → 0, to a Lorentzian metric g on ∂Ω × S1. When M = ∂Ω andθ is a positively oriented contact form on M then C(M) is trivial i.e.there is a principal bundle isomorphism Φ : C(M)→ M × S1 and theLorentzian metrics Fθ and g are conformally equivalent i.e. Φ∗g = efFθfor some f ∈ C∞(C(M)).

Fefferman’s metric is both an object worth of further investigationand, within the limits of our present knowledge of its properties, atool of first magnitude. By Fefferman’s own work, if M ⊂ Cn+1 isa strictly pseudoconvex real hypersurface then Chern-Moser’s chainsare the projections on M via π : C(M) → M of the non-vertical nullgeodesics of Fθ. This enabled L. Koch to give a new proof of the resultby H. Jacobowitz, that any two nearby points on M may be joined bya chain. While Koch’s proof of Jacobowitz’s theorem isn’t necessarilysimpler, it has the advantage of showing that there is an alternative,and very appealing, approach to CR and pseudohermitian geometrythrough Lorenzian geometry.

A piece of folklore is that just any nice problem in CR geometry be-gins with, or is related to, Fefferman’s metric. One illustrious instanceof that is CR Yamabe’s problem i.e. given a contact form θ ∈ P+ findu ∈ C∞(M) such that (the Tanaka-Webster connection associated tothe contact form) euθ has constant pseudohermitian scalar curvature:this turns out to be precisely the Yamabe problem for the Feffermanmetric Fθ. Relevant equation is certainly non-elliptic but its projectionon M turns out to be (nonlinear) subelliptic and then subelliptic tech-nicalities (whose basics were developed for the occasion) may be usedto prove existence of the solution. Fundamental work in this directionis due to D. Jerison and J.M. Lee, and to Gamara and Yacoub in smalldimension. The geometric interpretation of Jost and Xu’s subellipticharmonic maps belongs here (it relies on projecting on M the harmonicmaps system on C(M)).

By some of my older work, if M = S2n+1 then the first Pontryaginform of Fθ vanishes (i.e. P1(Ω2) = 0) and the corresponding trans-gression class is integral (i.e. [TP1(ω)] ∈ H3(L(C(M)),Z)). In generalPontyagin forms of the Fefferman metric are CR invariants, and if somePontryagin form vanishes then the corresponding transgression class isa CR invariant, as well. I understand there is a lot of progress in thisarea but, to my knowledge the characteristic rings of C(M) and Mhave not been effectively related. Fefferman’s metric (and correspond-ing Levi-Civita connection, which by the way may be very effectivelyrelated to the Tanaka-Webster connection) should be used to computethe Weil homomorphism w : I(GL(2n+ 2,R))→ H∗(C(M),R).

16

4. Bergman kernel and metric

Let Ω ⊂ Cn be a domain and L2(Ω) the Lebesgue space of all mea-surable functions f : Ω→ C such that

‖f‖L2 =

(∫Ω

|f(z)|2 dµ(z)

)1/2

<∞

where µ is the Lebesgue measure on R2n. Let

(f, g)L2 =

∫Ω

f(z) g(z) dµ(z)

be the inner product on L2(Ω). Let H2(Ω) be the space of all functionsf ∈ L2(Ω) which are holomorphic in Ω. Let A ⊂ Ω be a compactsubset and let f ∈ H2(Ω) an L2 holomorphic function on Ω. Let ζ ∈ Aand

P (ζ, ρ) = z ∈ Cn : |zj − ζj| < ρj , 1 ≤ j ≤ na polidisc of poliradius ρ = (ρ1, · · · , ρn) such that P (ζ, ρ) ⊂ Ω and fmay be represented as a uniformly convergent power series

f(z) =∞∑|α|=0

aα(z − ζ)α , z ∈ P (ζ, ρ).

Then

‖f(z)‖2L2 =

∫Ω

|f(z)|2 dµ(z) ≥∫P (ζ,ρ)

f(z) f(z) dµ(z) =

=∞∑|α|=0

∞∑|β|=0

aα aβ

∫P (ζ,ρ)

(z − ζ)α(z − ζ)β dµ(z) =

(as terms of the form aα(z − ζ)α and aβ(z − ζ)β with α 6= β aremutually orthogonal with respect to the L2 inner product (g, h)ρ =∫P (ζ,ρ)

g(z)h(z) dµ(z))

=∞∑|α|=0

∫P (ζ,ρ)

|aα|2n∏j=1

|zj − ζj|2αj dµ(z) ≥

≥∣∣a(0,··· ,0)

∣∣2 ∫P (ζ,ρ)

dµ(z) = |f(z)|2 µ (P (ζ, ρ))

that is|f(z)| ≤ µ (P (ζ, ρ))−1/2 ‖f‖L2 .

A compactness argument now shows that

(12) |f(z)| ≤ CA‖f‖L2

17

for some constant CA > 0 (depending only on the compact set A) andany f ∈ H2(Ω), z ∈ A. The simple estimate (12) plays a crucial role.First, it clearly means that for every z ∈ Ω the evaluation functional

δz : H2(Ω)→ C, δz(f) = f(z), f ∈ H2(Ω),

is continuous. As a second, and a bit less obvious, consequence of (12)the subspace H2(Ω) is closed in L2(Ω) and hence a Hilbert space itself.By the Riesz representation theorem for every z ∈ Ω there is an L2

holomorphic function kz ∈ H2(Ω) such that

δz(f) = (f , kz)L2 , f ∈ H2(Ω),

or

(13) f(z) =

∫Ω

K(z, ζ) f(ζ) dµ(ζ)

for any f ∈ H2(Ω) and any z ∈ Ω, where we have set K(z, ζ) = kz(ζ)for any z, ζ ∈ Ω. The function K : Ω× Ω → C is the Bergman kernelof Ω (and identity (13) show that K reproduces the L2 holomorphicfunctions).

If φνν≥0 is a complete orthonormal system in H2(Ω) then

K(z, ζ) =∞∑ν=0

φν(z)φν(ζ)

with uniform convergence on compact subsets Ω × Ω. It follows thatK : Ω× Ω→ C is a C∞ map, holomorphic in z and anti-holomorphicin ζ. Also if Ω is bounded than for each z0 ∈ Ω there is f ∈ H2(Ω)such that f(z0) 6= 0 hence the identity

K(z, z) =∞∑ν=0

|φν(z)|2

implies that K(z, z) > 0 for every z ∈ Ω. Consequently we may con-sider

g =n∑

j,k=1

∂2 logK(z, z)

∂zj ∂zkdzj dzk

which turns out to be a Kahlerian metric on Ω (the Bergman met-ric) with respect to which biholomorphisms of Ω are isometries (i.e.Hol(Ω) ⊂ Isom(Ω, g)). Also if Ω is homogeneous (i.e. Hol(Ω) is transi-tive on Ω) then (Ω, g) is Kahler-Einstein.

18

5. Fefferman’s asymptotic expansion formula for theBergman kernel and its consequences

If Ω = Bn = z ∈ Cn : |z| < 1 is the unit ball then its Bergmankernel is

K(z, ζ) =Cn(

1− z · ζ)n+1 , Cn =

n!

πn,

so its properties (e.g. smoothness at the boundary) can be read offdirectly from its expression. There are few other instances of domainsΩ ⊂ Cn where Bergman’s kernel may be explicitly computed. Howeverwhen Ω is a smoothly bounded strictly pseudoconvex domain then ourknowledge rests on a few classical results as follows. One such result isN. Kerzman’s theorem that K ∈ C∞(Ω×Ω\∆) where ∆ = (z, z) : z ∈∂Ω is the diagonal of the boundary. Proof is a wonderful applicationof the solution to the ∂-Neumann problem.

Another such result, with deep differential geometric implications aswe shall see, is Fefferrman asymptotic formula for the Bergman ker-nel. Precisely, again for any smoothly bounded strictly pseudoconvexdomain Ω = ρ < 0 ⊂ Cn

(14) K(ζ, z) = CΩ|∇ρ(z)|2 · detLρ(z) ·Ψ(ζ, z)−(n+1) + E(ζ, z)

where E ∈ C∞(Ω× Ω \∆) and

(15) |E(ζ, z)| ≤ C ′Ω|Ψ(ζ, z)|−(n+1)+1/2 · | log |Ψ(ζ, z)||.

Here Lρ is the Levi form and

Ψ(ζ, z) = [F (ζ, z)− ρ(z)]χ(|ζ − z|) + (1− χ(|ζ − z|)) |ζ − z|2 ,

F (ζ, z) = −n∑j=1

∂ρ

∂zj(z)(ζj − zj)− 1

2

n∑j,k=1

∂2ρ

∂zj ∂zk(z)(ζj − zj)(ζk − zk),

and χ(t) is a C∞ cut-off function with χ(t) = 1 for |t| < ε0/2 andχ(t) = 0 for |t| ≥ 3ε0/4. By (14)

K(z, z)−1/(n+1) = |ρ(z)|[Φ(z) + E(z, z) |ρ(z)|n+1

]−1/(n+1)

where Φ(z) ≡ CΩ|∇ρ(z)|2 detLρ(z) stays finite near ∂Ω and (by (15))

|E(z, z)| |ρ(z)|n+1 ≤ C ′Ω|ρ(z)|1/2| log |ρ(z)|| → 0, z → ∂Ω.

Therefore K(z, z)−1/(n+1) vanishes at ∂Ω. Also, as Φ(z) 6= 0 near ∂Ω,it may be shown that ∇K(z, z)−1/(n+1) 6= 0 along ∂Ω. Hence

ϕ(z) = −K(z, z)−1/(n+1)

19

is a defining function for Ω. Let

θ =i

2

(∂ − ∂

)ϕ, g =

∂2 logK(z, z)

∂zj ∂zkdzj dzk .

Mere differentiation shows that the Bergman metric g may be expressedas

(16) g(X, Y ) =n+ 1

ϕ

i

ϕ

(∂ϕ ∧ ∂ϕ

)(X, JY )− (dθ)(X, JY )

for any X, Y ∈ X(Ω), where J is the complex structure on Cn. For-mula (16) relates computationally the Kabler geometry of (Ω, g) tothe contact geometry of (∂Ω, θ) and this relationship has already beenexploited by A. Koranyi and H.M. Reimann in a beautiful paper (cf.[17]) where they relax the hypothesis of Fefferman’s theorem that bi-holomorphisms of smoothly bounded strictly pseudoconvex domainsextend smoothly at the boundary. If Φ : Ω → Ω is a boholomorphismthen its boundary values φ : ∂Ω→ ∂Ω must be a CR isomorphism andin particular a contact transformation (i.e. φ∗H(∂Ω) ⊂ φ−1H(∂Ω)).When Φ : Ω→ Ω is but a symplectomorphism (with respect to the sym-plectic structure −i ∂∂ logK(z, z) underlying the Kahlerian structureof (Ω, g)) extending smoothly at the boundary, the result by Koranyiand Reimann alluded to is that its boundary values φ : ∂Ω → ∂Ω areat least a contact transformation.

6. Boundary values of Bergman-harmonic maps

Let

Mδ = z ∈ Ω : ϕ(z) = −δ, δ > 0,

be the level sets of ϕ. For δ sufficiently small Mδ is still a smoothreal hypersurface, and a strictly pseudoconvex CR manifold for thatmatter. Therefore there is a one-sided neighborhood V ⊂ Ω of ∂Ωcarrying a foliation F such that

V/F = Mδ : 0 < δ ≤ δ0

for some δ0 > 0. Let H(F)→ V and T1,0(M)→ V be respectively thebundles whose portions over a leaf Mδ are the Levi distribution H(Mδ)and the CR structure T1,0(Mδ) of the leaf. One has

T1,0(F) ∩ T0,1(F) = (0), T0,1(F) ≡ T1,0(F),

Z,W ∈ C∞(T1,0(F)) =⇒ [Z,W ] ∈ C∞(T1,0(F)).

The rudiments of a general theory of tangentially CR foliations hasbeen already started (cf. e.g. S. Dragomir and S. Nishikawa, [7]). By

20

a result of J.M. Lee and R. Melrose there is a unique complex vectorfield ξ of type (1, 0) on V such that

∂ϕ(ξ) = 1, ∂∂ϕ (ξ, Z) = 0, Z ∈ T1,0(F).

The functionr = 2 ∂∂ϕ (ξ, ξ)

is the transverse curvature of F . The same result by J.M. Lee andR. Melrose (cf. [19]) shows that r is smooth up to the boundary i.e.r ∈ C∞(Ω). Let ξ = 1

2(N − iT ) be the real and imaginary parts of

ξ. Then T (ϕ) = 0 hence T ∈ T (F). Also T c dθ = 0 so that T istransverse to H(F) and actually

T (F) = H(F)⊕ RT.Let gθ be given by

gθ(X, Y ) = (dθ)(X, JY ), gθ(X,T ) = 0, gθ(T, T ) = 1.

Then gθ is a tangential Riemannian metric for F i.e. a Riemannianbundle metric on T (F) → V (so that the pullback of gθ to a leaf ofF is the Webster metric of the leaf). A rather obvious consequence of(16) is that the Bergman metric g of Ω and the tangential metric gθare related

(17) g(X, Y ) = −n+ 1

ϕgθ(X, Y ), X, Y ∈ H(F),

(18) g(X,T ) = 0, g(X,N) = 0, X ∈ H(F),

(19) g(T,N) = 0, g(T, T ) = g(N,N) =n+ 1

ϕ

(1

ϕ− r).

From the point of view of a differential geometer, the meaning (andusefulness) of formulae (17)-(19) is that one may relate, in an effectivemanner, linear connections on V parallelizing g to linear connectionson V parallelizing gθ. The linear connection parallelizing the Bergmanmetric one wishes to deal with is the Levi-Civita connection of (Ω, g).Indeed one wishes to study the Bergman-harmonic map equations

(20) −∆gΦi +(Γijk Φ

) ∂Φj

∂xA∂Φk

∂xBGAB = 0

whose principal part is the Bergman Laplacian

∆gu = −2n∑A=1

EA(EA(u))− (∇g

EAEA)(u)

, u ∈ C2(Ω),

which is expressed (by using a local g-orthonormal frame EA : 1 ≤A ≤ 2n of T (Ω)) in terms of covariant derivatives relative to ∇g, the

21

Levi-Civita connection of (Ω, g). As to linear connections parallelizinggθ, these are morally the Tanaka-Webster connections of the leaves ofF (since, as emphasized earlier, the pullback of gθ to a leaf of F isthe Webster metric of that leaf). Yet what one formally needs is alinear connection on V , rather that on some leaf of F . One is thus ledto the question whether one may piece together the Tanaka-Websterconnections of the leaves of F . That this is indeed feasible is a result byR. Graham and J.M. Lee (cf. [13]) producing a linear connection ∇ onV whose pointwise restriction to each leaf Mδ is the Tanaka-Websterconnection of Mδ. It is the unique linear connection ∇ on V (referredhereafter as the Graham-Lee connection) singled out by the followingaxioms i) T1,0(F) is parallel with respect to ∇, ii) ∇Lθ = 0, ∇T = 0,∇N = 0, and iii) the torsion T∇ is pure i.e.

T∇(Z,W ) = 0, T∇(Z,W ) = 2iLθ(Z,W )T,

T∇(N,W )rW + i τ(W ), Z,W ∈ T1,0(F),

τ (T1,0(F)) ⊂ T0,1(F), τ(N) = −J∇Hr − 2rT,

where

Lθ(Z,W ) = −i(dθ)(Z,W ), τ(X) = T∇(T,X), X ∈ T (V ).

Indeed (17)-(19) may then be used to relate ∇g to ∇. For instance

(21) ∇gXY = ∇XY+

+

ϕ

1− ϕrgθ(τX, Y ) + gθ(X,φY )

T−

−gθ(X, Y ) +

ϕ

1− ϕrgθ(X,φτY )

N

for any X, Y ∈ H(F). Here φ is defined by

φ : T (F)→ T (F),

φX = JX, φT = 0, X ∈ H(F).

Similar formulae may be got for ∇gXT , ∇g

XN , ∇gTN , ∇g

TT and ∇gNN

hence we express ∇g as a function of the Graham-Lee connection andthe transverse curvature of F and its first order derivatives

(22) ∇g = f(∇, r,∇Hr, T (r), N(r)

).

The whole point is that, when we shall analyze (20) as ϕ → 0 (equiv-alently as z → ∂Ω), the quantities gθ, ∇ and τ stay finite at theboundary and give there the Webster metric, the Tanaka-Webster con-nection, and the pseudohermitian torsion of the boundary. So does r

22

and its derivatives of any order, by the J.M. Lee and R. Melrose resultrecalled above. Formula (22) relies on the decomposition

(23) T (V ) = T (F)⊕ RN = H(F)⊕ RT ⊕ RN(as (21) already suggests). To exploit said decomposition one choosesa local orthonormal frame Wα : 1 ≤ α ≤ n − 1 of T1,0(F) i.e.gθ(Wα,Wβ) = δαβ and sets

Eα =

√− ϕ

n+ 1Wα , En =

√2ψϕ

n+ 1ξ, ψ ≡ ϕ

1− rϕ.

Then Ej : 1 ≤ j ≤ n is a local orthonormal frame of T 1,0(V ) i.e.g(Ej, Ek) = δjk, adapted to the decomposition (23) and (22) yields

(24) ∆g = − ϕ

n+ 1∆b −

2ϕ(n− 1)

n+ 1N+

+ψϕ

n+ 1

N2 + T 2 +∇Hr + 2rN

where ∆b is given by

∆b =n−1∑α=1

(WαWα +WαWα −∇WαWα −∇WαWα) .

(25)(Γijk Φ

) ∂Φj

∂xA∂Φk

∂xBGAB =

= − 2ϕ

n+ 1

(Γijk Φ

)n−1∑α=1

Wα(Φj)Wα(Φk)− 2ψ ξ(Φj) ξ(Φk)

.

Taking into account (24)-(25) the Bergman-harmonic map system (20)may be written

(26) ∆bφi + 2(n− 1)Nφi − ψ

(N2 + T 2 +∇Hr + 2rN

)φi+

+2(Γijk φ

)n−1∑α=1

Wα(φj)Wα(φk)− 2ψξ(φj)ξ(φk)

= 0.

Let φ = φf be the solution to the Dirichlet problem for the system (20)with the boundary condition φ = f on ∂Ω with f ∈ C∞(∂Ω, S). Letus assume that φ ∈ C∞(Ω, S). Then as ϕ→ 0 the equation (26) leadsto

(HSf)i + 2(n− 1)Nf i = 0, 1 ≤ i ≤ ν.

The normal derivatives of the map f : ∂Ω→ S may thus be determinedin terms of purely tangential quantities, a phenomenon looked upon astypically non-elliptic. We close with the statement (cf. [5])

23

Theorem 2. Let Ω ⊂ Cn (n ≥ 2) be a smoothly bounded strictlypseudoconvex domain and g the Bergman metric on Ω. Let S be acomplete ν-dimensional (ν ≥ 2) Riemannian manifold of sectionalcurvature Sect(S) ≤ κ2 for some κ > 0. Assume that S may becovered by one coordinate chart χ = (y1, · · · , yν) : S → Rν. Letf ∈ W 1,2(Ω, S) ∩ C0(Ω, S) be a map such that f(Ω) ⊂ B(p, µ) forsome p ∈ S and some 0 < µ < minπ/(2κ) , i(p) where i(p) is theinjectivity radius of p. Let φ = φf : Ω → S be the solution to theDirichlet problem

(27) τg(φ) = 0 in Ω, φ = f on ∂Ω.

If f ∈ C∞(∂Ω , S) and φ ∈ C∞(Ω, S

)then

(28) N(f i) = − 1

2(n− 1)(HSf)i , 1 ≤ i ≤ ν,

for any local coordinate system (ω, yi) on S such that φ(Ω) ∩ ω 6= ∅(f i = yi f). Also N = −JT and T is the characteristic directionof ∂Ω. In particular if N(f i) = 0 then f : ∂Ω → S is a subellipticharmonic map.

References

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