Uniqueness, symmetry, and embeddedness of minimal surfaces
type: Item
instance of: scholarly article
title: Uniqueness, symmetry, and embeddedness of minimal surfaces
zbMATH Open document ID: 0575.53037
published in: Journal of Differential Geometry
publication date: 1983-01-01T00:00:00Z
review text: \textit{M. Shiffman} [Ann. Math., II. Ser. 63, 77-90 (1956; Zbl 0070.168)] proved that an immersed minimal surface M in \(R^ 3\) of genus zero with \(\partial M=\Gamma_ 1\cup \Gamma_ 2\), where \(\Gamma_ 1\) and \(\Gamma_ 2\) are convex curves lying in parallel planes, meets each intermediate plane transversally in a convex curve. In the first part of this paper the author extends this result into various directions. For example the topological restriction on M is removed and the ambient space may have higher dimension. In addition to this the (nontrivial) problem is discussed under which assumptions symmetries of the boundary are inherited by the spanning minimal hypersurface. Details are too complicated to be given here. The second part of this paper introduces minimal hypersurfaces which are 'regular at infinity'. It is shown that reasonable behavior of the normal vectors at infinity implies this property. Finally the author proves that any complete minimal hypersurface which is regular at infinity and has two ends is a catenoid or a pair of planes.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 49Q05
zbMATH DE Number: 3919253
zbMATH Keywords: Plateau problem
zbMATH Keywords: minimal hypersurface
zbMATH Keywords: regular at infinity
zbMATH Keywords: normal vectors
zbMATH Keywords: catenoid
author: Richard Schoen
reviewed by: Bernd Wegner
MaRDI profile type: Publication
Wikidata QID: Q115187891
full work available at URL: https://doi.org/10.4310/jdg/1214438183
OpenAlex ID: W1567841374
DOI: 10.4310/JDG/1214438183
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description: scientific article; zbMATH DE number 3919253