Stability of area-preserving variations in space forms
type: Item
instance of: scholarly article
title: Stability of area-preserving variations in space forms
zbMATH Open document ID: 1156.53037
author: Yijun He
author: Haizhong Li
published in: Annals of Global Analysis and Geometry
publication date: 2008-07-30T00:00:00Z
review text: The authors study compact hypersurfaces \(M\) of real space forms. They define inductively the functional \(\mathcal A_r = \int_M F_r dM\), where \(F_0=1\), \(F_1=S_1\), \(F_r = S_r+c(n-r+1)/(r-1) F_{r-2}\) and \(S_r\) denotes the r-th elementary symmetric function associated to the shape operator of the immersion. They consider the problem of minimizing \(\mathcal A_r\) while keeping the area of \(M\) constant and show that critical points of this variational problem are hypersurfaces for which the quotient of \(S_{r+1}\) and \(S_1\) is a constant. It is also shown that, in case \(S_1\) is positive and the surrounding space is the Euclidean space, a critical immersion is stable if and only if \(M\) is a sphere immersed in the Euclidean space as a totally umbilical hypersurface. Similar results are also obtained in case \(M\) is an hypersurface contained in the hyperbolic space or contained in an open hemisphere of the unit sphere.
reviewed by: Luc Vrancken
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A30
Mathematics Subject Classification ID: 53C43
zbMATH DE Number: 5306112
zbMATH Keywords: stability
zbMATH Keywords: variation problems
zbMATH Keywords: area-preserving variation
zbMATH Keywords: higher order mean curvatures
MaRDI profile type: Publication
full work available at URL: s10455-007-9095-3
OpenAlex ID: W2123264923
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DOI: 10.1007/S10455-007-9095-3
description: scientific article; zbMATH DE number 5306112