On the unique continuation problem for CR mappings into nonminimal hypersurfaces
type: Item
instance of: scholarly article
title: On the unique continuation problem for CR mappings into nonminimal hypersurfaces
zbMATH Open document ID: 0914.32005
DOI: 10.1007/BF02921657
author: Peter Ebenfelt
published in: The Journal of Geometric Analysis
publication date: 1999-06-28T00:00:00Z
review text: Let \(M\) and \(M'\) be germs at 0 of \(C^\infty\) real hypersurfaces of \(\mathbb{C}^{n+1}\) and \(\mathbb{C}^{n'+1}\) respectively. As such we can assume \(M\) is given by \(Im w= \Phi(Z,\overline Z), Rew)\) for a \(C^\infty\) function \(\Phi: \mathbb{C}^{n+1}\to\mathbb{R}\) in coordinates \((Z,w)\) for \(\mathbb{C}^n\times\mathbb{C}\), and \(M'\) is similarly represented by \(\Phi':\mathbb{C}^{n'+1}\to\mathbb{R}\) in coordinates \((Z', w')\) for \(\mathbb{C}^{n'}\times \mathbb{C}\). Let \(H=(f_1, \dots, f_{n'},g)\) be a CR function from \(M\) to \(M'\). The author is interested in the question of whether infinite order of vanishing of \(H\) at 0 \((H\) ``flat'' at 0) forces \(H\) to be identically 0. The author obtains the following results. If \(M'\) contains a holomorphic hypersurface (Hence \((Z',w')\) can be choosen so that \(\Sigma'= \{w'= 0\}\subset M'.)\), \(H\) extends holomorphically to one side of \(M\), and \(H(M) \not\subset\Sigma'\) (Hence \(g\not\equiv 0.)\), then \(g\) is not flat at 0. (Hence \(H\) is not flat.) The author points out that any CR map \(f:M\to\mathbb{C}^{n'}\) yields a CR map \(H=(f,0):M\to M'\) which can be flat but not identically 0, hence the necessity of the hypothesis \(g\not\equiv 0\). The author continues the investigation without the hypothesis that \(H\) extends holomorphically to one side of \(M\). In case \(M'\) is Levi flat he shows there is a nonconstant CR mapping \(H:M\to M'\) which is flat at the origin. Further he provides an example of non-Levi flat \(M\) and \(M'\) in \(\mathbb{C}^2\) for which such a CR mapping exists. Finally in case \(M\) and \(M'\) are real analytic the author obtains sufficient condition for \(H\) flat to imply \(H\) constant. There conditions involve \(M\) being ``sufficiently far from being Levi-flat''.
Mathematics Subject Classification ID: 32V40
Mathematics Subject Classification ID: 32V99
zbMATH DE Number: 1160126
zbMATH Keywords: smooth hypersurface
zbMATH Keywords: CR mapping
zbMATH Keywords: order of vanishing
reviewed by: G. A. Harris
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1160126