SMOOTHNESS OF CAUCHY RIEMANN MAPS FOR A CLASS OF REAL HYPERSURFACES

We study the regularity problem for Cauchy Riemann maps between hypersurfaces in Cn. We prove that a continuous Cauchy Riemann map between two smooth C∞ pseudoconvex decoupled hypersurfaces of finite D’Angelo type is of class C∞. Introduction Many classical problems in complex analysis rely on the boundary behavior of holomorphic maps and, as a consequence, on the regularity of Cauchy Riemann maps between real hypersurfaces. Only partial results have been obtained when the hypersurfaces are not assumed real analytic: the smoothness of a continuous Cauchy Riemann (CR) map between smooth real hypersurfaces was proved, for instance, in [17] for strictly pseudoconvex hypersurfaces, in [11] for pseudoconvex hypersurfaces of finite D’Angelo type in C, and in [8] for a Lipschitz CR map between convex hypersurfaces of finite D’Angelo type. It is natural to study decoupled hypersurfaces (or domains) to understand the link between complex dimension two and higher complex dimensions: for such domains J. D. McNeal [16] gave estimates on Bergman, Caratheodory and Kobayashi metrics, J. E. Fornæss and J. D. McNeal [14] constructed local peak holomorphic functions at boundary points, and D. C. Chang and S. Grellier [7] gave properties of the Szegö projection under an additive global assumption. In this paper we prove the following local result: Theorem 1. Let Γ1 and Γ2 be two C∞ pseudoconvex real hypersurfaces in C, containing the origin 0, and let f be a continuous non constant CR map from Γ1 to Γ2, satisfying f(0) = 0. If Γ1 and Γ2 are decoupled, of finite D’Angelo type at the origin, then f is a smooth C∞ locally finite map near the origin. 2000 Mathematics Subject Classification. Primary: 32H40, 32H99; Secondary: 32F40, 32G07, 32H15, 32H35, 32M99.


Introduction
Many classical problems in complex analysis rely on the boundary behavior of holomorphic maps and, as a consequence, on the regularity of Cauchy Riemann maps between real hypersurfaces.Only partial results have been obtained when the hypersurfaces are not assumed real analytic: the smoothness of a continuous Cauchy Riemann (CR) map between smooth real hypersurfaces was proved, for instance, in [17] for strictly pseudoconvex hypersurfaces, in [11] for pseudoconvex hypersurfaces of finite D'Angelo type in C 2 , and in [8] for a Lipschitz CR map between convex hypersurfaces of finite D'Angelo type.It is natural to study decoupled hypersurfaces (or domains) to understand the link between complex dimension two and higher complex dimensions: for such domains J. D. McNeal [16] gave estimates on Bergman, Caratheodory and Kobayashi metrics, J. E. Fornaess and J. D. McNeal [14] constructed local peak holomorphic functions at boundary points, and D. C. Chang and S. Grellier [7] gave properties of the Szegö projection under an additive global assumption.
In this paper we prove the following local result: Theorem 1.Let Γ 1 and Γ 2 be two C ∞ pseudoconvex real hypersurfaces in C n+1 , containing the origin 0, and let f be a continuous non constant CR map from Γ 1 to Γ 2 , satisfying f (0) = 0.If Γ 1 and Γ 2 are decoupled, of finite D'Angelo type at the origin, then f is a smooth C ∞ locally finite map near the origin.
According to a result of S. Bell and D. Catlin [4], it is sufficient to prove that the set f −1 (f (0)) is compact near the origin i.e. that for any neighborhood U of 0 the set f −1 (f (0)) ∩ U is relatively compact in U .Using a dilation we reduce the study of the regularity of f to the study of the boundary behavior of a holomorphic map F between rigid algebraic domains D 1 and D 2 .This map is the limit of some holomorphic maps F ν .The algebraicity of domains D 1 and D 2 gives some information on the map F , detailed in Proposition 1.2.Moreover one can prove that if the dilated maps F ν satisfy uniform Hölder estimates then the non compactness of the set f −1 (f (0)) implies the non compactness of F −1 (F (0)) (the origin is mapped to the origin by the scaling process applied to Γ 1 ).Such Hölder estimates may be obtained when the sequence (F ν (0)) ν is bounded; we prove this property on (F ν (0)) ν , and consequently Theorem 1, as soon as the hypersurface ∂D 1 is not spherical, i.e. not locally biholomorphic to a sphere at a strictly pseudoconvex point.This relies on a classification of vector fields tangent to a weighted homogeneous rigid polynomial hypersurface, given by E. Bedford and S. Pinchuk [3], and on a precise study of the restrictions of F to some subvarieties of D 1 (Sections 2 and 3).If ∂D 1 is spherical it may happen that lim z∈D1 z→0 |F (z)| = ∞.Using the characterization of spherical rigid hypersurfaces, given by A. V. Isaev [15] and obtained in the spirit of the Chern-Moser theory, we describe F in terms of a correspondence of the unit ball.This proves the compactness of F −1 (∞) in ∂D 1 and ends the proof of Theorem 1 (Section 4).We note that these techniques also provide a modified proof of Theorem 0.1 in [11].

Reduction of the problem
Decoupled hypersurfaces are defined as follows: Definition 1.1.A hypersurface Γ in C n+1 containing the origin 0 is decoupled at 0 if there are a neighborhood U of 0, holomorphic coordinates (z 0 , z) = (z 0 , z 1 , . . ., z n ) centered at 0 and a defining function r such that: where f j is a real function for every j = 1, . . ., n.
Let Γ 1 = {(z 0 , z) ∈ U : r 1 (z 0 , z) = 0} and Γ 2 = {(z 0 , z) ∈ U : r 2 (z 0 , z) = 0} be two smooth C ∞ decoupled pseudoconvex hypersurfaces in C n+1 , of finite type 2m and 2k respectively (in the sense of D'Angelo [12]) at the origin.We may assume that the functions f 1 j and f 2 j , in the expansions of r 1 and r 2 , are subharmonic functions of class C ∞ , without harmonic terms, vanishing at order less than or equal to 2m and 2k respectively.If f is a continuous CR map from Γ 1 to Γ 2 then according to [4] we have: Since the order of contact between the (j + 1)th coordinate complex line and Γ 1 at the origin is less than or equal to 2m we may write for every 1 ≤ j ≤ n: f 1 j (z j , z j ) = H j (z j , z j ) + R j (z j , z j ) where H j is a homogeneous polynomial of degree 2m j ≤ 2m and R j denotes terms of larger degree.If (p ν ) ν is a sequence of points in Ω 1 converging to 0 then (f (p ν )) ν = (q ν ) ν converges to 0 by (ii).We may assume that there is a unique point z ν ∈ Γ 2 such that dist(q ν , Γ 2 ) = |q ν − z ν | = δ ν .Let L 0 , . . ., L n be the vector fields defined by L 0 = ∂ ∂z 0 and for 1 ≤ j ≤ n, span the real tangent space to Γ 2 and {L 1 , . . ., L n } span CT 1,0 (Γ 2 ).For every ν, let L j s,t r 2 (z ν ) be the commutator of L j , L j of length s − 1 in z j and t − 1 in z j at z ν .Since Ω 2 is of finite type we can set: M j = inf{m j such that L j s,t r 2 (z ν ) = 0 for some s, t such that s + t = m j }, and define the dilation: Let us consider automorphisms U ν of C n+1 converging to the identity with U ν (q ν ) = (−δ ν , 0).We may choose U ν such that r 2 • (U ν ) −1 is decoupled and has no harmonic terms in z 1 , . . ., z n .
If p ν = (−ε ν , 0) is on the real inward normal to Ω 1 at the origin and Without any restriction we may assume that each map F ν is defined on Ω ν 1 with values in Ω ν 2 where Ω ν We obtain after extraction of subsequences: Proposition 1.1.
(i) The sequence (Ω ν 1 ) ν converges, in the Hausdorff convergence, to the domain where P j is a non harmonic subharmonic polynomial of degree less than or equal to 2k. (iii) The family (F ν ) ν converges uniformly on compact subsets of D 1 to a holomorphic map Proof of Proposition 1.1:The expression of the dilation Λ ν 1 gives part (i).
Part (ii): by the definition of τ ν j there is a real positive constant C such that for every sufficiently large ν and every j ≥ 2, τ ν j ) where P j is a subharmonic polynomial of degree less than or equal to 2k, without harmonic term.Part (iii) was proved by F. Berteloot [5] in According to the Gauss formula we have for every fixed 1 ≤ j ≤ n: where hk = (h k 1 , . . ., h k n ) and ∆ t = {λ ∈ C : |λ| < t}.Using the expression of r ν 2 we get: 2 ) j is the restriction of r ν 2 to the jth complex coordinate axis.However for every 0 < r < 1 we have: Hence, since r ν 2 • hk (e iθ ) ≤ − Re h k 0 (e iθ ), we obtain: • hk (0) being bounded from above independently of k, the sequence (h k j ) k converges uniformly on compact subsets of ∆ to a holomorphic map h from ∆ to D 2 .Since (h k (0)) k is relatively compact in D 2 we have the inclusion h(∆) ⊂ D 2 .By covering the unit ball of C n+1 by disks we get the same convergence for any sequence (h k ) k of holomorphic maps defined on the unit ball and also for any sequence of holomorphic maps defined on D 1 .This proves part (iii).
It happens that the map F has some intrinsic properties coming mainly from the rigidity of the domains D 1 and D 2 .For instance we know by [18] that if, in our situation, there is a sequence There exist strictly pseudoconvex points p in ∂D 1 and q in ∂D 2 such that F extends to a biholomorphism in a neighborhood of p with F (p) = q.(iv) F is algebraic.
The tools used to prove Proposition 1.2, developed in different papers and valid for decoupled domains, are the existence of peak plurisubharmonic functions with algebraic growth (constructed in [13]) implying an equivalence distance property for f (part (i)), and the existence of peak holomorphic functions at infinity for rigid domains, given by [2] (part (iii)).Part (ii) uses the transversality of the Segre varieties, satisfied in any dimension and part (iv) is an immediate consequence of part (iii) by [19].The complete proof of Proposition 1.2 is given by Proposition 1.5 and Lemma 2.2 of [8].
Without changing the properties of D 2 , we may assume that q = 0.The map F inherits some properties from f , since the family (F ν ) ν satisfies the following uniform properties: Proposition 1.3.Let a be a point in ∂D 1 and (a ν ) ν a sequence of points in ∂Ω ν 1 , converging to a. (i) If (F ν (a ν )) ν is bounded then there exists a neighborhood U of a in C n+1 and a real positive constant C such that for every positive integer ν and every z, z in Proof: See Propositions 4.1 and 5.1 of [10].
We deduce from Propositions 1.2 and 1.3 the following result: Proposition 1.4.If f satisfies the assumptions of Theorem 1 then the set f −1 (f (0)) is compact near the origin in Γ 1 and f is a C ∞ locally finite map at the origin under one of the following conditions: Proof of Proposition 1.4: Assume by contradiction that the set f −1 (f (0)) is not compact near the origin.Then there is, for every sufficiently small real positive number r, a point in f −1 (f (0)) of modulus r.Using the expression of the dilation we can find for every ν, k larger than or equal to one a point a ν k in ∂Ω ν 1 satisfying: We may assume that for every k the sequence (a ν k ) ν converges to a point a k in ∂D 1 with modulus 1/k: the sequence (a k ) k converges to 0 in ∂D 1 .
Part (ii): the non compactness of f −1 (f (0)) implies that lim ν→∞ |F ν (a ν k )| = +∞ for every k and thus by Proposition 1.3 part (ii) that the point a k belongs to F −1 (∞) for every k: this contradicts the finiteness of F −1 (∞).The smoothness of f is then given by [4].
According to Proposition 1.4 one needs to answer the two following questions to prove Theorem 1: In the next section we describe the CR infinitesimal automorphisms of ∂D 1 .We use this description in Section 3 to answer question 1.The answer to question 2 is given in Section 4.

Classification of vector fields
Since we deal with decoupled real hypersurfaces we start this section with some results on real hypersurfaces in complex dimension two.Let H be a real subharmonic non harmonic homogeneous polynomial defined in C, and Γ be the real hypersurface defined by: Γ = {(z 0 , z 1 ) ∈ C 2 : Im z 0 + H(z 1 , z 1 ) = 0}.The real dimension of the real vector space of CR infinitesimal automorphisms at a strictly pseudoconvex point of Γ is equal to 2, 3 or 8 according to a result of E. Cartan [6].This equals 8 if Γ is spherical, in which case we may assume that H(z 1 , z 1 ) = |z 1 | 2m , and 3 if Γ is a tube hypersurface (and We assume in this section that the hypersurface ∂D 1 has at least one non spherical direction at point p given by Proposition 1.2.This means that if we write p = (p 0 , . . ., p n ) then there is an integer l satisfying 0 ≤ l ≤ n − 1 such that for every integer j larger than l the hypersurface Γ j = {(z 0 , z j ) ∈ C 2 : Im z 0 + H j (z j , z j ) = − k =j H k (p k , p k } is not biholomorphic to the unit sphere at (p 0 , p j ).In the following l denotes the smallest integer satisfying this condition and we remark that the equality l = 0 means that ∂D 1 is a spherical hypersurface in C n+1 .The infinitesimal CR automorphisms of ∂D 1 are then given by the following proposition: • Case 2: l > 0.
There exist (n +2) real constants a 0 , b 0 , b 1 , . . ., b n and l(l − 1) complex constants c jk defined for 1 ≤ j, k ≤ l with j = k such that: Proof of Proposition 2.1: Since case 1 may be considered as a special case of case 2, let us study case 2. We may assume that for every integer k between 1 and l the polynomial H k is defined for z k in C by: Let j be an integer between l + 1 and n.Fixing all the variables z k = p k for k different from j (1 ≤ k ≤ n) the vector field where j z = (z 0 , p 1 , . . ., p j−1 , z j , p j+1 , . . ., p n ), is tangent to the hypersurface    at (p 0 , p j ).As S j is not spherical, every local tangent vector field extends to a global one acording to [6].Then, because of the homogeneity of S j , each homogeneous part in the expansion of X j is also a CR infinitesimal automorphism for S j .Since S j is of finite type the real vector space of such vector fields has a finite dimension; thus X j is a polynomial vector field which means that all the coefficients of X j are polynomial.Hence the vector field X defined for (z 0 , z l+1 , . . ., z n ) in a neighborhood of (p 0 , p l+1 , . . ., p n ) in C n−l+1 by: where z l = (z 0 , p 1 , . . ., p l , z l+1 , . . ., z n ), is polynomial.Let us expand X in homogeneous vector fields with respect to the weights 2m l+1 , . . ., 2m n (we recall that 2m j is the degree of polynomial H j given by Proposition 1.1) with the convention that ∂/∂z j has weight −1/2m j .The classification of homogeneous vector fields Q tan- is given by [3] (Lemmas 2.7, 3.4, 3.5).Since there is no spherical direction in S the admissible weights for these vector fields are −1 (corresponding to the translation ∂ ∂z0 ), −1/2m j (with l + 1 ≤ j ≤ n) and 0. Moreover the only CR infinitesimal automorphism of weight 0 corresponds to the dilation: A straightforward computation based on Lemma 2.7 of [3] gives the form of the homogeneous CR infinitesimal automorphisms of weight −1/2m j and consequently of the CR infinitesimal automorphisms of weight wt(Q) with −1 < wt(Q) < 0: Thus we have X = X0 where a 0 , b 0 , b l+1 , . . ., b n are real analytic real functions of the variables z 1 , . . ., z l , defined locally at (p 1 , . . ., p l ).Thus the vector field X satisfies the following system: for (z 0 , z) close to p. Since the vector field X is holomorphic it follows from the second equation of system (S) that b 0 , b l+1 , . . ., b n are real constants.
Let us fix an integer j such that 1 ≤ j ≤ l.
Differentiating the first equation of system (S) with respect to z j we have: Since X j is holomorphic there exist: • A holomorphic function f j of the variables z 1 , . . ., z l such that f j |{zj =0} = 0.
• A holomorphic function fj depending on the variables z k for 1 ≤ k ≤ l, k = j.
• A real analytic function g j not depending on z j such that a 0 = z j mj (f j + fj ) + g j .
Since a 0 is a real function and f j is holomorphic we have f j (z 1 , . . .,z l )= where α j is a real constant and g j = z mj j fj + gj where gj is a real analytic real function depending neither on z j nor on z j .Then a 0 = α j |z j | 2mj + (z j mj fj + z mj j fj ) + gj .Replacing X 1 , . . ., X l in terms of ∂ j a 0 (see equation ( 1)) in the first equation of system (S) we obtain the following equality: and adding these l equalities for 1 ≤ j ≤ l: Let us identify the holomorphic terms in this expression.Since l k=1 z m k k fk is not holomorphic, this is a real function and we have: It is then sufficient to set b j = α j /m j and c jk = a jk /m j for 1

Classification of maps from D 1 to D 2
We recall that the limit map F obtained in Section 1 is locally proper according to Proposition 1.2 part (i).We assume in this section that ∂D 1 has at least one non spherical direction and we use the description of CR infinitesimal automorphisms given by Proposition 2.1 to obtain the following: Proof of Proposition 3.1: According to Proposition 1.2 parts (iii)-(iv) the map G = F −1 , locally defined at 0 with G(0) = p, is an algebraic map.Since the vector field G ( ∂ ∂z 0 ) is a CR infinitesimal automorphism at p the map G satisfies: • In case 2 of Proposition 2.1: Since system (S1) can be considered as a subsystem of system (S2) we will focus on the resolution of system (S2).The map G being algebraic the resolution of the equation Moreover there exist functions G0 , . . ., Gn , Gjk (1 ≤ j ≤ l, 0 ≤ k ≤ l−1), holomorphic in a neighborhood U of 0 in C n , such that: Let us write for every integer j in {1, . . ., n} H j (z j + p j , z j + p j ) = Q j (z j , z j ) + Im(S j (z j , z j )) + H j (p j , p j ) where Q j is a real subharmonic polynomial without harmonic term and S j is a holomorphic polynomial without constant term.If T is the transformation: Let us denote T • G = g = (g 0 , . . ., g n ).By assumption g(0) = 0.For every z 0 in U the function g is well defined on the half plane Λ • g is the identity in a neighborhood of the origin on Λ z 0 .
Assume now that either Gjk (z 0 ) = 0 for some (j, k) with 1 ≤ j ≤ l, 0 ≤ k ≤ l − 1 or b j = 0 for some j with l + 1 ≤ j ≤ n.According to system (S3) the restriction of g to Λ z 0 is a polynomial map of the variable z 0 and the restriction of the function Im g 0 + n j=1 Q j (g j , g j ) to Λ z 0 is a negative subharmonic polynomial function.Its weighted is subharmonic on C and its Laplacian has an algebraic growth; this is a subharmonic polynomial.So there are an integer k 0 and complex constants α s,t , defined for (s, t) with 1 ≤ s, t ≤ k 0 , such that: Considering this as a polynomial in fj (ζ k ) and since P j is not identically zero, there is an integer s between 1 and 2k such that d k0+1 dz k0+1 (( fj (z k )) s ) is identically zero on C: the map z k → ( fj (z k )) s is a polynomial.Hence the holomorphic function z k → fj (z k ), globally defined on C, is polynomial and thus the map F • T is polynomial.Finally the point F (0) = (F •T )(T −1 (0)) is a finite point in ∂D 2 .This proves Proposition 3.1.

The end of the proof of Theorem 1
We proved in Section 3 that if ∂D 1 has at least one non spherical direction then F (0) is a finite point.Theorem 1 is then a consequence of Proposition 1.4.Assume now that the hypersurface ∂D 1 is spherical at point p.We may write: D 1 = {(z 0 , z) ∈ C n+1 : Im z 0 + n j=1 |z j | 2mj < 0}.By Proposition 1.2 part (iii) the domain D 2 is spherical at the origin.Hence by a work of A. V. Isaev [15] the polynomial P = n j=1 P j satisfies a system of partial differential equations: where E l , D l j , C l jk and H jk are holomorphic functions defined in a neighborhood of the origin.It follows by setting z k = 0 for k = j that the polynomial P j satisfies an equivalent system and the domain {(z 0 , z j ) ∈ C 2 : Im z 0 +P j (z j , z j ) < 0} is also spherical in C 2 for every j in {1, . . ., n} by Proposition 2.2 of [15].Since P j has no pure anti-holomorphic term we obtain by a comparison of the terms of maximal degree in z j that E j = D j j = H jj = 0. Thus P j satisfies the following system in C 2 : ∂ 2 P j ∂z 2 j (z j , z j ) = C j jj (z j ) ∂P j ∂z j (z j , z j ) where C j jj is a holomorphic function.An integration shows that there exists a holomorphic polynomial h j such that P j (z j , z j ) = |h j (z j )| 2 .Since D 1 is biholomorphic to the ellipsoïd {(z 0 , z) ∈ C n+1 : |z 0 | 2 + n j=1 |z j | 2mj < 1} one can prove as in Proposition 3.1 of [8] that the map F is a proper map from D 1 to D 2 .Thus if g 1 and g 2 are the maps defined in C n+1 by g 1 : (z 0 , z) → (z 0 , z m1 1 , . . ., z mn n ) and g 2 : (z 0 , z) → (z 0 , h 1 (z 1 ), . . ., h n (z n )), there is a proper holomorphic auto correspondence F of the unbounded representation H of the unit ball of C n+1 such that: F • g 1 = g 2 • F ( [1]).Every component of its graph is the graph of an automorphism of H. Since F −1 (∞) is contained in ( F •g 1 ) −1 (∞) and every automorphism of H extends to a homeomorphism from H ∪ {∞} onto H ∪ {∞} we obtain that F −1 (∞) is compact in ∂D 1 .Theorem 1 is then given by Proposition 1.4 condition (ii).