Local Holomorphic Equivalence of Real Analytic Submanifolds in ℂ^N

Summary

This paper presents some recent results of the authors jointly with Peter Ebenfelt concerning local biholomorphisms which map one realanalytic or real-algebraic submanifold of ℂ^N into another. It is shown that under some optimal conditions such mappings are determined by their jets of a predetermined finite order at a given point. Under these conditions, if the manifolds are algebraic, it is also shown that the components of the holomorphic mappings must be algebraic functions. The stability group of self mappings is shown to be a finite dimensional Lie group for most points in the case of real-analytic holomorphically nondegenerate real hypersurfaces in ℂ^N. The notion of Segre sets associated to a point of a real-analytic CR submanifold of ℂ^N is one of the main ingredients in this work. Properties of these sets and their relationship to minimality of these manifolds are discussed.

Introduction

We consider here some recent results concerning local biholomorphisms which map one real analytic (or real algebraic) subset of ℂ^N

into another such subset of the same dimension. One of the general questions studied is the following. Given M, M’ c ℂ^N, germs of real analytic subsets at p and p’ respectively with diniR M = dim_ℝ M', describe the (possibly empty) set of germs of biholomorphisms H : (ℂ^N,p) → (ℂ^N,p') with H(M) ⊂ M'.

Most of the new results stated here have been recently obtained in joint work with Peter Ebenfelt. We shall give precise definitions and specific references in the text.