Hypersurfaces with constant mean curvature and prescribed area
type: Item
instance of: scholarly article
title: Hypersurfaces with constant mean curvature and prescribed area
zbMATH Open document ID: 0945.49030
DOI: 10.1007/BF02567956
author: Frank Duzaar
published in: Manuscripta Mathematica
publication date: 2000-10-03T00:00:00Z
full work available at URL: https://eudml.org/doc/156231
review text: Fixing an \(n\)-dimensional rectifiable current \(T_0\) in \(\mathbb{R}^{n+1}\) having rectifiable boundary \(B=\partial T_0\), the author considers the family of rectifiable currents \(T\) with \(\partial T= B\). It is assumed that the support of \(B\) has \(n\)-dimensional Hausdorff measure \(0\). The volume enclosed by \(T-T_0\) is defined to be \(\langle Q,\Omega\rangle\), where \(Q\) is the unique \((n+1)\)-dimensional current with finite mass and boundary \(T-T_0\), and where \(\Omega\) is the volume form on \(\mathbb{R}^{n+1}\). By maximizing the volume enclosed by \(T-T_0\), subject to a constraint on the mass \(\mathbb{M}(T)\), the author proves the existence of surfaces with boundary \(B\) and having constant mean curvature off their possible singular sets of Hausdorff dimension \(n-7\). The bulk of the paper is then devoted to considering how the mean curvature of these maximizing surfaces depends on the mass constraint on \(T\). The main result is that, as the mass constraint approaches infinity, the mean curvature behaves as would be expected for a sphere.
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 58E12
Mathematics Subject Classification ID: 53C65
zbMATH DE Number: 980732
zbMATH Keywords: rectifiable currents
zbMATH Keywords: mean curvature
zbMATH Keywords: maximizing surfaces
zbMATH Keywords: mass constraint
reviewed by: Harold Parks
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 980732