On the analyticity of smooth CR mappings between real-analytic CR manifolds
type: Item
instance of: scholarly article
title: On the analyticity of smooth CR mappings between real-analytic CR manifolds
zbMATH Open document ID: 0997.32033
author: Sylvain Damour
published in: Michigan Mathematical Journal
publication date: 2002-05-20T00:00:00Z
review text: The main theorem of this article provides a synthesis of a great number of results about the analyticity of smooth CR mappings, starting with the classical Lewy-Pinchuk principle. It also provides an entirely new proof of the Baouendi-Jacobowitz-Trèves analyticity theorem [\textit{M. S. Baouendi, H. Jacobowitz} and \textit{F. Trèves}, Ann. Math. (2) 122, 365-400 (1985; Zbl 0583.32021)], without any rank condition on the mapping. As a matter of fact, essentially all smooth reflection principles obtained by Baouendi-Rothschild and others are covered by the main theorem of the article under review. The general framework of this work was introduced in two previous articles by \textit{B. Coupet, S. Pinchuk} and \textit{A. Sukhov} [C. R. Acad. Sci., Paris Sér. I, Math. 329, No. 6, 489-494 (1999; Zbl 0949.32018) and Math. Z. 235, No. 3, 541-557 (2000; Zbl 0972.32008)] where the problem is dealt with for a codimension one source CR manifold. Let \(f: M\to M'\) be a \({\mathcal C}^\infty\)-smooth CR mapping between two real analytic generic submanifolds of \({\mathbb C}^n\) and \({\mathbb C}^{n'}\). Let \(p\in M\) and assume that \(M\) is minimal in the sense of Trépreau-Tumanov at \(p\), so CR functions extend holomorphically to a wedge at \(p\). Let \(\overline{L}_1,\dots,\overline{L}_m\) be a basis of \((0,1)\) vector fields tangent to \(M\), where \(m=\text{ CRdim} M\). Define the first characteristic variety associated with \(f\) to be \[ {\mathbf V}_p':=\{z'\in {\mathbb C}^{n'}:\overline L^\alpha \rho_k'(z', \overline{f(\cdot)}) |_p=0, \forall \alpha\in {\mathbb N}^m, \forall k=1, \dots,d'\}, \] where \(\rho_k'(z',\bar z')=0\), \(k=1,\dots,d'\) are defining equations for \(M'\) and where \(p'=f(p)\). We have \(p'\in {\mathbf V}_p'\). The main result states that if the dimension of \({\mathbf V}_p'\) is zero at \(p'\), then \(f\) extends holomorphically to a neighborhood of \(p\). Two main ideas are used in the proof. The first one is to deal abstractly with some fields…
reviewed by: Joël Merker
Mathematics Subject Classification ID: 32V25
Mathematics Subject Classification ID: 32D10
Mathematics Subject Classification ID: 32V35
Mathematics Subject Classification ID: 32V10
Mathematics Subject Classification ID: 32V40
zbMATH DE Number: 1742562
zbMATH Keywords: CR manifolds
zbMATH Keywords: CR mappings
zbMATH Keywords: minimality in the sense of Trépreau-Tumanov
zbMATH Keywords: reflection principle
zbMATH Keywords: envelope of meromorphy
zbMATH Keywords: direction of wedge extendability
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1012409972
OpenAlex ID: W2046319818
DOI: 10.1307/MMJ/1012409972
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description: scientific article; zbMATH DE number 1742562