CR maps and point Lie transformations
type: Item
instance of: scholarly article
title: CR maps and point Lie transformations
zbMATH Open document ID: 1024.32019
published in: Michigan Mathematical Journal
publication date: 2003-06-30T00:00:00Z
review text: The motivation of the paper under review lies in the following well-known finite jet determination result which follows from the work of \textit{S. S. Chern} and \textit{J. K. Moser} [Acta Math. 133, 219-271 (1974; Zbl 0302.32015)]. Real-analytic CR automorphisms of Levi-nondegenerate real-analytic hypersurfaces in complex space are determined by their 2-jets at any given point. This kind of uniqueness result in CR geometry has been generalized to much wider classes of real-analytic hypersurfaces (and even CR manifolds of arbitrary codimension) by \textit{M. S. Baouendi}, \textit{L. P. Rothschild} and the reviewer [J. Geom. Anal. 12, No. 4, 543-580 (2002)]. In the paper under review, the author proposes a new approach in deriving Chern-Moser's uniqueness result by studying Lie symmetries of certain holomorphic second-order partial differential equations. Indeed, by a fundamental observation due to \textit{S. Webster} [Invent. Math. 43, 53-68 (1977; Zbl 0348.32005)], Segre varieties are invariant under biholomorphic mappings and hence such mappings can be seen as Lie symmetries of associated second-order partial differential equation (PDE) systems. The finite jet determination result of Chern-Moser can then be seen as a consequence of a 2-jet determination property for a certain class of PDE systems (Theorem 1.1). The well-known above-mentioned uniqueness result seems however the only application to CR geometry that the author can offer in the paper under review.
Mathematics Subject Classification ID: 32V40
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 35A30
zbMATH DE Number: 1943226
zbMATH Keywords: Levi-nondegenerate real-analytic hypersurfaces
zbMATH Keywords: Lie symmetries
author: Alexandre Sukhov
reviewed by: Nordine Mir
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1028575739
OpenAlex ID: W2086153258
DOI: 10.1307/MMJ/1028575739
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description: scientific article; zbMATH DE number 1943226