Stable hypersurfaces with constant scalar curvature
type: Item
instance of: scholarly article
title: Stable hypersurfaces with constant scalar curvature
zbMATH Open document ID: 0792.53057
DOI: 10.1007/BF03025712
author: Hilário Alencar
author: Manfredo P. do Carmo
author: Antonio Gervasio Colares
published in: Mathematische Zeitschrift
publication date: 1994-08-02T00:00:00Z
full work available at URL: https://eudml.org/doc/174520
review text: It is well known that hypersurfaces with constant mean curvature in Riemannian spaces are solutions to the variational problem of minimizing area for volume-preserving variations. Less known is the fact that hypersurfaces with constant scalar curvature in Riemannian spaces of constant sectional curvature are also solutions to a variational problem, namely, that of minimizing the integral of the mean curvature for volume preserving variations. Thus questions of stability can be posed for hypersurfaces with constant scalar curvature. The following result is proved. Let \(M\) be a compact immersed hypersurface of the euclidean space, or of a sphere, with constant scalar curvature. In the case the ambient space is a sphere, assume, in addition, that \(M\) is contained in an open hemisphere. Then \(M\) is stable if and only if \(M\) is a geodesic sphere. The question envolves the study of a second order differential operator related to the Laplacian and the second fundamental form of the hypersurface.
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 549783
zbMATH Keywords: variational problem
zbMATH Keywords: volume preserving variations
zbMATH Keywords: stability
reviewed by: Manfredo P. do Carmo
MaRDI profile type: Publication
cites work: Stability of hypersurfaces with constant mean curvature
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cites work: Hypersurfaces with constant scalar curvature
cites work: Some integral formulas for closed hypersurfaces
cites work: Q3210491
cites work: Variational properties of functions of the mean curvatures for hypersurfaces in space forms
OpenAlex ID: W3217470764
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description: scientific article; zbMATH DE number 549783