Levi-flat extensions from a part of the boundary.
type: Item
instance of: scholarly article
title: Levi-flat extensions from a part of the boundary.
zbMATH Open document ID: 1042.32015
author: Nikolay Shcherbina
author: Giuseppe Tomassini
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2004-01-28T00:00:00Z
review text: Let \(G\) be a bounded domain in \(\mathbb{C}\times\mathbb{R}\) such that \(G\times i\mathbb{R} \subset\mathbb{C}^2\) is strictly pseudoconvex, and let \(U\) be open subset of boundary \(bG\). The authors show that there is an open subset \(\Omega^U\) of \(\overline G\) with \(\Omega^U\cap bG=U\) such that for every \(\varphi\in C(U)\) there exist two functions \(\Phi^\pm\in C(\Omega^U)\) with \(\Phi^\pm| _U=\varphi\) such that the graphs \(\Gamma(\Phi^\pm)\) (over \(\Omega^U\cap G\)) are Levi-flat, and such that \(\Phi^-\leq\Phi\leq\Phi^+\), if \(\Phi\in C(\Omega^U)\), \(\Phi| _U=\varphi\), and its graph \(\Gamma(\Phi)\) is Levi-flat. The authors also show that \(\Omega^U\) is the maximal domain of Levi-flat extensions of the graph of some continuous function on \(U\), when \(G\) is diffeomorphic to a \(3\)-ball and \(U\) satisfies some topological conditions. A sketch of the proof of the main result is given.
reviewed by: Xianghong Gong
Mathematics Subject Classification ID: 32V25
Mathematics Subject Classification ID: 32D15
Mathematics Subject Classification ID: 32T15
Mathematics Subject Classification ID: 32E20
zbMATH DE Number: 2034208
zbMATH Keywords: Levi-flat extension
zbMATH Keywords: strictly pseudoconvex domain
MaRDI profile type: Publication
full work available at URL: j.crma.2003.10.015
OpenAlex ID: W1978480093
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DOI: 10.1016/J.CRMA.2003.10.015
description: scientific article; zbMATH DE number 2034208