On Levi-flat hypersurfaces with given boundary in \(\mathbb C^n\)
type: Item
instance of: scholarly article
title: On Levi-flat hypersurfaces with given boundary in \(\mathbb C^n\)
zbMATH Open document ID: 1151.32013
author: Pierre Dolbeault
published in: Science in China. Series A
publication date: 2008-09-09T00:00:00Z
review text: The author discusses the following problem. Consider a compact connected \(2\)-codimensional submanifold \(S\subset {\mathbb{C}}^n \) and find a Levi flat hypersurface \(M\subset {\mathbb{C}}^n\setminus S\) such that \(dM=S\) i.e. \(M\) has the boundary \(S\)(possibly as current). The author studies the behaviour at \(CR\)-points, at complex points and elliptic or hyperbolic flat points. Then he studies the induced foliation by \(CR\)-orbits in the case of elliptic flat points and in the case of hyperbolic flat points. He considers the boundary value problem for embedded surfaces, studying the subvarieties with negligible singularities and treating the same boundary problem in a real hyperplane of \({\mathbb{C}}^n\). Finally, he presents the problem of Levi flat \((2n-1)\)-subvarieties with a given boundary in \({\mathbb{C}}^n\). The case where \(n=2\) is considered separately.
reviewed by: Vasile Oproiu
Mathematics Subject Classification ID: 32V40
zbMATH DE Number: 5323338
zbMATH Keywords: Levi flat hypersurfaces
zbMATH Keywords: \(CR\)-submanifolds, complex manifolds
MaRDI profile type: Publication
full work available at URL: s11425-007-0200-7
OpenAlex ID: W1993102083
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DOI: 10.1007/S11425-007-0200-7
description: scientific article; zbMATH DE number 5323338