Dmitri Zaitsev

Equivalence and embedding problems for CR-structures of any codimension

An approach is suggested to equivalence and embedding problems for smooth CR-submanifolds of complex spaces (and, more generally, for abstract CR-manifolds) in terms of complete differential systems in jet bundles satisfied by all CR-equivalences or CR-embeddings respectively. For equivalence problems, manifolds are assumed to be of finite type and finitely nondegenerate. These are higher order generalizations of the corresponding nondegeneracy conditions for the Levi form. It is shown by a simple example that these nondegeneracy conditions cannot be even slightly relaxedto more general known conditions. In particular, for essentially finite hypersurfaces in C 2 , such a complete system does not exist in general. For embedding problems, source manifolds are assumed to be of finite type and their embeddings to be finitely nondegenerate. Sufficient conditions on CR-manifolds are given, where the last condition is automatically satisfied by all CR-embeddings. 2005 Elsevier Ltd. All rights reserved.

Approximation and convergence of formal CR-mappings

An important step in understanding the existence of analytic objectswith certain properties consists of understanding the same problem at the level of formal power series. The latter problem can be reduced to a sequence of algebraic equations for the coefficients of the unknown power series and is often simpler than the original problem, where the power series are required to be convergent. It is therefore of interest to know whether such power series are automatically convergent or can possibly be replaced by other convergent power series satisfying the same properties. A celebrated result of this kind is Artin’s approximation theorem [1] which states that a formal solution of a system of analytic equations can be replaced by a convergent solution of the same system that approximates the original solution at any prescribed order. In this paper, we study convergence and approximation properties (in the spirit of [1]) of formal (holomorphic) mappings sending real-analytic submanifolds M ⊂ C and M ′ ⊂ C ′ into each other, N,N ′ ≥ 2. In this situation, the above theorem of Artin cannot be applied directly. Moreover, without additional assumptions on the submanifolds, the analogous approximation statement is not even true. Indeed, in view of an example of Moser-Webster [23], there exist real-algebraic surfaces M,M ′ ⊂ C that are formally but not biholomorphically equivalent. However, our firstmain result shows that this phenomenon cannot happen if M is a minimal CR-submanifold (not necessarily algebraic) in C (see Section 2.1 for notation and definitions). Theorem 1.1. Let M ⊂ C be a real-analytic minimal CR-submanifold and M ′ ⊂ C ′ a real-algebraic subset with p ∈ M and p ′ ∈ M ′. Then for any formal (holomorphic)

On local CR-transformations of Levi-degenerate group orbits in compact Hermitian symmetric spaces

We present a large class of homogeneous

Dynamics of one-resonant biholomorphisms

2

Obstructions to embeddability into hyperquadrics and explicit examples

-nondegenerate CR-manifolds

Rigidity of CR maps between Shilov boundaries of bounded symmetric domains

M

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

, both of hypersurface type and of arbitrarily high CR-codimension, with the following property: Every CR-equivalence between domains

Extension of CR-functions into weighted wedges through families of nonsmooth analytic discs

U,V

Non-embeddable real algebraic hypersurfaces

in

Formal and finite order equivalences

M