Convergence of formal embeddings between real-analytic hypersurfaces in codimension one

We show that every formal embedding sending a real-analytic strongly pseudoconvex hypersurface in M\subset \C^N into another such hypersurface in M'\subset \C^{N+1} is convergent. More generally, if $M$ and $M'$ are merely Levi-nondegenerate, the same conclusion holds for any formal embedding provided either that the embedding is CR transversal or the target hypersurface does not contain any complex curves.Comment: 8 page

Formal biholomorphic maps of real analytic hypersurfaces

Let f : (M,p)\to (M',p') be a formal biholomorphic mapping between two germs of real analytic hypersurfaces in \C^n, p'=f(p). Assuming the source manifold to be minimal at p, we prove the convergence of the so-called reflection function associated to f. As a consequence, we derive the convergence of formal biholomorphisms between real analytic minimal holomorphically nondegenerate hypersurfaces. Related results on partial convergence of formal biholomorphisms are also obtained.Comment: 15 pages, Late

Analytic regularity of CR maps into spheres

Let $M$ be a connected real-analytic hypersurface in \C^N and $\S$ the unit real sphere in \C^{N'}, $N'> N\geq 2$. Assume that $M$ does not contain any complex-analytic hypersurface of \C^N and that there exists at least one strongly pseudoconvex point on $M$. We show that any CR map $f\colon M\to \S$ of class $C^{N'-N+1}$ extends holomorphically to a neighborhood of $M$ in \C^N.Comment: 11 page

Algebraic approximation in CR geometry

We prove the following CR version of Artin's approximation theorem for holomorphic mappings between real-algebraic sets in complex space. Let M\subset \C^N be a real-algebraic CR submanifold whose CR orbits are all of the same dimension. Then for every point $p\in M$, for every real-algebraic subset S'\subset \C^N\times\C^{N'} and every positive integer $\ell$, if f\colon (\C^N,p)\to \C^{N'} is a germ of a holomorphic map such that {\rm Graph}\, f \cap (M\times \C^{N'})\subset S', then there exists a germ of a complex-algebraic map f^\ell \colon (\C^N,p)\to \C^{N'} such that {\rm Graph}\, f^\ell \cap (M\times \C^{N'})\subset S' and that agrees with $f$ at $p$ up to order $\ell$.Comment: To appear in J. Math. Pures App

Finite jet determination of local CR automorphisms through resolution of degeneracies

Let M be a connected real-analytic hypersurface in N-dimensional complex euclidean space whose Levi form is nondegenerate at some point. We prove that for every point p in M, there exists an integer k=k(M,p) such that germs at p of local real-analytic CR automorphisms of M are uniquely determined by their k-jets (at p). To prove this result we develop a new technique that can be seen as a resolution of the degeneracies of M. This procedure consists of blowing up M near an arbitrary point p in M regardless of its minimality or nonminimality; then, thanks to the blow-up, the original problem can be reduced to an analogous one for a very special class of nonminimal hypersurfaces for which one may use known techniques to prove the finite jet determination property of its CR automorphisms.Comment: 16 page

Parametrization of local CR automorphisms by finite jets and applications

For any real-analytic hypersurface M in complex euclidean space of dimension >= 2 which does not contain any complex-analytic subvariety of positive dimension, we show that for every point p in M the local real-analytic CR automorphisms of M fixing p can be parametrized real-analytically by their l(p)-jets at p. As a direct application, we derive a Lie group structure for the topological group Aut(M,p). Furthermore, we also show that the order l(p) of the jet space in which the group Aut(M,p) embeds can be chosen to depend upper-semicontinuously on p. As a first consequence, it follows that that given any compact real-analytic hypersurface M in complex euclidean space, there exists an integer k depending only on M such that for every point p in M germs at p of CR diffeomorphisms mapping M into another real-analytic hypersurface in a complex space of the same dimension are uniquely determined by their k-jet at that point. Another consequence is a boundary version of H. Cartan's uniqueness theorem. Our parametrization theorem also holds for the stability group of any essentially finite minimal real-analytic CR manifold of arbitrary codimension. One of the new main tools developed in the paper, which may be of independent interest, is a parametrization theorem for invertible solutions of a certain kind of singular analytic equations, which roughly speaking consists of inverting certain families of parametrized maps with singularities.Comment: to appear in J. Amer. Math. So

Approximation and convergence of formal CR-mappings

Let $M\subset C^N$ be a minimal real-analytic CR-submanifold and $M'\subset C^{N'}$ a real-algebraic subset through points $p\in M$ and $p'\in M'$. We show that that any formal (holomorphic) mapping $f\colon (C^N,p)\to (C^{N'},p')$, 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 $E'\subset M'$ of points of D'Angelo infinite type. Therefore, if $M'$ does not contain any nontrivial complex-analytic subvariety through $p'$, any formal map $f$ as above is necessarily convergent