Local holomorphic foliations and partial analyticity of \(C^\infty\) CR mappings.
type: Item
instance of: scholarly article
title: Local holomorphic foliations and partial analyticity of \(C^\infty\) CR mappings.
zbMATH Open document ID: 1029.32017
author: Sylvain Damour
published in: Manuscripta Mathematica
publication date: 2002-12-01T00:00:00Z
review text: Consider a smooth CR mapping \(f\) between a generic real analytic submanifold \(M \subset \mathbb{C}^{n}, n>1\), and a real analytic subset \(M' \subset \mathbb{C}^{n'}\). Let \(p \in M, p'=f(p)\), and \(\tau (f)\) be the germ at \((p,p')\) of a smaller complex analytic subset of \(\mathbb{C}^{n+n'}\) containing a graph of \(f\) in a neighborhood of \((p,p')\). Partial analytic degree of \(f\) at \(p\), denoted by \(\deg_{p}f\), is defined to be the integer \(\dim T_{p}(f)-n\), and \(M'\) is said to be \((r,s)\)-flat at \(p\), if it contains a real analytic submanifold of dimension \(r\) passing the point \(p'\) and biholomorphic to the Cartesian product \(N \times D\), where \(N \subset \mathbb{C}^{\nu}\), \(\nu \geq 0\), is a real analytic submanifold and \(D \subset \mathbb{C}^{s}\) is a domain. Here the author establishes an upper estimate of the partial analyticity of \(f\) in terms of the maximal dimension of the local holomorphic foliations contained in \(M'\), which is the main theorem of this paper: Let \(f: M \to M'\) be a smooth CR mapping between a submanifold \(M \subset \mathbb{C}^{n}\), which is real analytic and minimal at \(p \in M\), and a real analytic subset \(M' \subset \mathbb{C}^{n'}\). If \(\deg_{z}f\) is a constant equal to \(s\) in a neighborhood of \(p\), then \(M'\) is \((r,s)\)-flat at \(f(q)\), where \(r \geq \operatorname {rank} \max_{p}f\), for each point \(q\) of an open dense subset of \(M\) near to \(p\).
reviewed by: Na Jisheng
Mathematics Subject Classification ID: 32V25
Mathematics Subject Classification ID: 32V40
Mathematics Subject Classification ID: 32H99
zbMATH DE Number: 1837445
zbMATH Keywords: CR mapping
zbMATH Keywords: generic real analytic submanifold
zbMATH Keywords: local holomorphic foliation
zbMATH Keywords: partial analyticity
MaRDI profile type: Publication
full work available at URL: s00229-002-0300-1
OpenAlex ID: W2332998509
DOI: 10.1007/S00229-002-0300-1
description: scientific article; zbMATH DE number 1837445