Real-analytic CR Submanifolds Research Articles

Unique jet determination of CR maps into Nash sets

Let M⊂CN be a real-analytic CR submanifold, M′⊂CN′ a Nash set and EM′ the set of points in M′ of D'Angelo infinite type. We show that if M is minimal, then, for every point p∈M, and for every pair of germs of C∞-smooth CR maps f,g:(M,p)→M′, there exists an integer k=kp such that if f and g have the same k-jets at p, and do not send M into EM′, then necessarily f=g. Furthermore, the map p↦kp may be chosen to be bounded on compact subsets of M. As a consequence, we derive the finite jet determination property for pairs of germs of CR maps from minimal real-analytic CR submanifolds in CN into Nash subsets in CN′ of D'Angelo finite type, for arbitrary N,N′≥2.

Formal versus analytic CR mappings

We discuss the convergence/divergence problem for formal holomorphic mappings sending real-analytic CR submanifolds into real-analytic sets both lying in complex Euclidean spaces of arbitrary dimension. In particular, we survey the recent developments on

Divergent CR-equivalences and meromorphic differential equations

Using the analytic theory of differential equations, we construct, in any positive CR-dimension and CR-codimension, examples of formally but not holomorphically equivalent real-analytic CR-submanifolds in complex space.

Holomorphic Deformations of Real-Analytic CR Maps and Analytic Regularity of CR Mappings

Let \(M\subset {\mathbb {C}}^N\) and \(M'\subset {\mathbb {C}}^{N'}\) be real-analytic CR submanifolds, with M minimal. We provide a new sufficient condition, that happens to be also essentially necessary, for all sufficiently smooth CR maps \(h:U\rightarrow M'\) defined on a connected open subset of M and of rank larger than a prescribed integer r to be real-analytic on a dense open subset of U. This condition corresponds to the nonexistence of nontrivial holomorphic deformations of germs of real-analytic CR mappings whose rank is larger than r. As a consequence, we obtain several new results about analyticity of CR mappings that, at the same time, generalize and unify a number of previous existing ones.

Regularization of Local CR-Automorphisms of Real-Analytic CR-Manifolds

Let M be a connected generic real-analytic CR-submanifold of a finite-dimensional complex vector space E. Suppose that for every a∈M the Lie algebra \({\frak{hol}}(M,a)\) of germs of all infinitesimal real-analytic CR-automorphisms of M at a is finite-dimensional and its complexification contains all constant vector fields \(\alpha\frac {\partial}{\partial z} \), α∈E, and the Euler vector field \(z\frac{\partial}{\partial z} \). Under these assumptions we show that: (I) every \({\frak{hol}}(M,a)\) consists of polynomial vector fields, hence coincides with the Lie algebra \({\frak{hol}}(M)\) of all infinitesimal real-analytic CR-automorphisms of M, (II) every local real-analytic CR-automorphism of M extends to a birational transformation of E, and (III) the group Bir (M) generated by such birational transformations is realized as a group of projective transformations upon embedding E as a Zariski open subset into a projective algebraic variety. Under additional assumptions the group Bir (M) is shown to have the structure of a Lie group with at most countably many connected components and Lie algebra \({\frak{hol}}(M)\). All of the above results apply, for instance, to Levi non-degenerate quadrics, as well as a large number of Levi degenerate tube manifolds.

A note on CR mappings of positive codimension

We prove the following Artin type approximation theorem for smooth CR mappings: given M a connected real-analytic CR submanifold in C^N that is minimal at some point, M' a real-analytic subset of C^N', and H:M->M' a smooth CR mapping, there exists a dense open subset O in M such that for any q in O and any positive integer k there exists a germ at q of a real-analytic CR mapping H^k:(M,q)->M' whose k-jet at q agrees with that of H up to order k.

On the stability group of CR manifolds

For any essentially finite minimal real-analytic generic submanifold M ⊂ C N , N ⩾ 2 , we show that for every point p ∈ M the local real-analytic CR automorphisms of M fixing p can be parametrized real-analytically by their ℓ = ℓ ( p ) jets at p. As an application, we derive a Lie group structure for the stability group Aut ( M , p ) . We also show that the order ℓ = ℓ ( p ) of the jet space in which the group Aut ( M , p ) embeds can be chosen to depend upper-semicontinuously on p. This yields that given any compact real-analytic minimal CR submanifold M in C N , there exists an integer k depending only on M such that for every point p ∈ M local CR diffeomorphisms mapping a neighbourhood of p in M into another real-analytic CR submanifold in C N with the same CR dimension as that of M are uniquely determined by their k-jet at p. To cite this article: B. Lamel, N. Mir, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

On some rigidity properties of mappings between CR-submanifolds in complex space

We survey some recent results on holomorphic or formal mappings sending real submanifolds in complex space into each other. More specifically, the approximation and convergence properties of formal CR-mappings between real-analytic CR-submanifolds will be discussed, as well as the corresponding questions in the category of real-algebraic CR-submanifolds.

Holomorphic extension of smooth CR-mappings between real-analytic and real-algebraic CR-manifolds

The classical Schwarz reflection principle states that a continuous map f between real-analytic curves M and M ′ in C that locally extends holomorphically to one side of M , extends also holomorphically to a neighborhood of M in C. It is well-known that the higher-dimensional analog of this statement for maps f : M → M ′ between real-analytic CR-submanifolds M ⊂ C andM ′ ⊂ C ′ does not hold without additional assumptions (unlessM andM ′ are totally real). In this paper, we assume that f is C-smooth and that the target M ′ is real-algebraic, i.e. contained in a real-algebraic subset of the same dimension. If f is known to be locally holomorphically extendible to one side of M (when M is a hypersurface) or to a wedge with edge M (when M is a generic submanifold of higher codimension), then f automatically satisfies the tangential CauchyRiemann equations, i.e. it is CR. On the other hand, if M is minimal, any CR-map f : M → M ′ locally extends holomorphically to a wedge with edge M by Tumanov’s theorem [Tu88] and hence, in that case, the extension assumption can be replaced by assuming f to be CR. Local holomorphic extension of a CR-map f : M → M ′ may clearly fail when M ′ contains an (irreducible) complex-analytic subvariety E ′ of positive dimension and f(M) ⊂ E . Indeed, any nonextendible CR-function on M composed with a nontrivial holomorphic map from a disc in C into E ′ yields a counterexample. Our first result shows that this is essentially the only exception. Denote by E ′ the set of all points p ∈ M ′ through which there exist irreducible complex-analytic subvarieties of M ′ of positive dimension. We prove: Theorem 1.1. Let M ⊂ C and M ′ ⊂ C ′ be connected real-analytic and real-algebraic CRsubmanifolds respectively. Assume that M is minimal at a point p ∈ M . Then for any C-smooth CR-map f : M → M , at least one of the following conditions holds: (i) f extends holomorphically to a neighborhood of p in C ; (ii) f sends a neighborhood of p in M into E . If M ′ is a real-analytic hypersurface, the set E ′ consists exactly of those points that are not of finite type in the sense of D’Angelo [D82] (see Lempert [L86] for the proof) and, in particular,

Approximation and convergence properties of formal CR-maps

Let M⊂ C N be a minimal real-analytic CR-submanifold and M′⊂ C N′ a real-algebraic subset through points p∈ M and p′∈ M′ respectively. We show that that any formal (holomorphic) mapping f:( C N,p)→( C N′,p′) , sending M into M′, can be approximated up to any given order at p by a convergent map sending M into M′. If M is furthermore generic, we also show that any such map f, that is not convergent, must send (in an appropriate sense) M into the set E′⊂M′ of points of D'Angelo infinite type. Therefore, if M′ does not contain any nontrivial complex-analytic subvariety through p′, any formal map f sending M into M′ is necessarily convergent. To cite this article: F. Meylan et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 671–676.

Analytic regularity of CR-mappings

In this paper we give general conditions that guarantee the analyticity of C∞-smooth CR-mappings between real-analytic CR-submanifolds M ⊂ C and M ′ ⊂ C ′ (see Theorem 1.1). The proof of the analyticity is based on results on holomorphic and meromorphic extension of functions on wedge-like domains to boundary points (see Theorems 2.4 and 2.6) that may be of independent interest. The analyticity problem for CR-mappings f : M → M ′ between real-analytic CR-submanifolds goes back to the work of Lewy [Le56, Le77] and Pinchuk [Pi75], where the analyticity of a CR-diffeomorphism f was shown for M, M ′ ⊂

On dimensions of the groups of biholomorphic automorphisms of real-analytic hypersurfaces

In this paper we study the dependence of the local geometry of real-analytic hypersuffaces in ℂ n on the dimension of the group of biholomorphic automorphisms of this surface. We also classify the hypersurfaces in terms of this group. We present some examples showing that the classes of the given construction are not empty. We find a new formulation of the Freeman theorem on the so-called straightening of a real-analytic CR-submanifold in ℂ n with degenerate Levi form of constant rank.

Germs of local automorphisms of real-analytic CR structures and analytic dependence on $k$-jets

The topic of the paper is the study of germs of local holomor- phisms f between C n and C n ' such that f(M) � Mand df(T c M) = T c M ' for MC n and M ' � C n ' generic real-analytic CR submanifolds of arbitrary codimensions. It is proved that for M minimal and Mfinitely nondegener- ate, such germs depend analytically on their jets. As a corollary, an analytic structure on the set of all germs of this type is obtained. ' be connected locally closed real-analytic submanifolds, x ∈ M,x ' ∈ Mbe arbitrary points. The complex tangent subspace TxM ∩iTxM will be denoted by T c x M. M is a CR manifold, if dimT c x M is constant. In this case dimCRM := dimCT c M is called the CR dimension and codimCRM := dimIRTM − dimIRT c M the CR codimension. A CR submanifold M ⊂ C n is called generic, if TM +iTM = C n . Suppose for the moment that M,M ' ⊂ C n are generic real-analytic CR subman- ifolds of the same CR dimension and the same CR codimension. Baouendi, Ebenfelt and Rothschild found optimal nondegeneracy conditions on M and Msuch that a germ at x of a local biholomorphism f (between some neighborhoods in C n ) with f(M) ⊂ M ' ,f(x) = x ' , is uniquely determined by itsk-jetj k xf, wherek is an integer