Bernhard Lamel

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. Cartans 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.

Finite jet determination of CR embeddings

We prove finite jet determination results for smooth CR embeddings, which are of constant degeneracy, using the method of complete systems. As an application, we obtain a reflection principle for mappings between a Levinondegenerate hypersurface in ℂNand a Levinondegenerate hypersurface in ℂN+1.We also give an independent proof of the reflection principle for mappings between strictly pseudoconvex hypersurfaces in any codimension due to Forstneric [14].

Automorphism groups of minimal real-analytic CR manifolds

We show that the local automorphism group of a minimal real-analytic CR manifold M is a finite dimensional Lie group if (and only if) M is holomorphically nondegenerate by constructing a jet parametrization.

Remarks on the rank properties of formal CR maps

We prove several new transversality results for formal CR maps between formal real hypersurfaces in complex space. Both cases of finite and infinite type hypersurfaces are tackled in this note.

Lie group structures on automorphism groups of real-analytic CR manifolds

Given any real-analytic CR manifold

New extension phenomena for solutions of tangential Cauchy–Riemann equations

M

ASYMPTOTIC APPROXIMATIONS AND A BOREL-TYPE RESULT FOR CR FUNCTIONS

, we provide general conditions on

Explicit bounds for the finite jet determination problem

M

On the vanishing rate of smooth CR functions

guaranteeing that the group of all its global real-analytic CR automorphisms Aut

Regularity of infinitesimal CR automorphisms

_{CR}(M)