Embedding compact three-dimensional CR manifolds of finite type in \({\mathbb{C}}^ n\)
type: Item
instance of: scholarly article
title: Embedding compact three-dimensional CR manifolds of finite type in \({\mathbb{C}}^ n\)
zbMATH Open document ID: 0668.32019
DOI: 10.2307/1971491
author: Michael Christ
published in: Annals of Mathematics. Second Series
publication date: 1989-01-01T00:00:00Z
review text: The author proves that any smooth, compact, three dimensional CR manifold, X, which is pseudoconvex and of finite type, and on which \(\overline{\partial}_ b\) has closed range in \(L^ 2\) admits a CR embedding into some \({\mathbb{C}}^ n\). The author refers to the literature for the corresponding result in case X is strongly pseudoconvex, in this paper he is able to replace the hypothesis of strong pseudoconvexity with that of pseudoconvexity plus finite type. In order to obtain CR functions which separate points, the author constructs a one parameter family of CR functions all equal to one at a given point on X and tending to O elsewhere. He does this first for certain model hypersurfaces in \({\mathbb{C}}^ 2\) of the form \(M=\{(Z_ 1,Z_ 2):\) \(Im(Z_ 2)=P(Z_ 1)\}\), for a real valued homogeneous polynomial P. Near a point on X he shows how to choose coordinates so as to exhibit X as a perturbation of such a model. He then derives suitable CR functions on X via an approximation argument, the main tool being the \(L^ p\) Sobolev and Hölder regularity estimates for \(\overline{\partial}_ b\) developed by \textit{C. L. Fefferman} and \textit{J. J. Kohn} [Adv. Math. 69, 223-303 (1988; Zbl 0649.35068)] and by himself [J. Am. Math. Soc. 1, No.3, 587-646 (1988)].
Mathematics Subject Classification ID: 32V40
Mathematics Subject Classification ID: 35N15
Mathematics Subject Classification ID: 32W05
zbMATH DE Number: 4093883
zbMATH Keywords: CR manifold
zbMATH Keywords: pseudoconvex
zbMATH Keywords: finite type
zbMATH Keywords: CR embedding
zbMATH Keywords: CR functions
reviewed by: G. A. Harris
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971491
OpenAlex ID: W2335215912
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description: scientific article; zbMATH DE number 4093883