A characterization of Clifford tori with constant scalar curvature one by the first stability eigenvalue
type: Item
instance of: scholarly article
title: A characterization of Clifford tori with constant scalar curvature one by the first stability eigenvalue
zbMATH Open document ID: 1068.53042
author: Luis J. Alías
author: L. A. M. jun. Sousa
author: Aldir jun. Brasil
published in: Bulletin of the Brazilian Mathematical Society. New Series
publication date: 2005-02-11T00:00:00Z
review text: An immersion of an \(n\)-dimensional manifold M into the \((n+1)\)-dimensional sphere with constant scalar curvature one, under the metric induced by the sphere, is considered. \(M\) is taken to be a closed orientable manifold. Such surfaces can be characterized variationally as critical points of the integral of the mean curvature \(H\) over \(M\). A sharp upper bound for the first eigenvalue of the corresponding Jacobi operator is given in terms of \(H\), and it is shown that the upper bound is attained only by the Clifford tori in the \((n+1)\)-dimensional sphere. It is noted that the Jacobi operator is elliptic if and only if \(n\) is at least 3 and the third elementary symmetric function of the principal curvatures does not vanish, thus these two conditions are assumed. It then follows that \(H\) can be taken to be positive. Once the Jacobi operator is established to be elliptic, a min-max characterization of its first eigenvalue, using the Rayleigh quotient, can be applied. The primary result of the paper is that the first eigenvalue is less than or equal to \(-2n(n-1)\) times the minimum of \(H\), with equality holding only for the Clifford tori. This work is motived by similar results for minimal hypersurfaces in spheres by Simons, Chern do Carmo, Kobayashi, Lawson, Perdomo, Wu and Santos. The authors dedicate this paper to the memory of Jose F. Escobar.
reviewed by: Wayne Rossman
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 2133667
zbMATH Keywords: scalar curvature one
zbMATH Keywords: hypersurface
zbMATH Keywords: spherical space
zbMATH Keywords: Jacobi operator
zbMATH Keywords: first eigenvalue
zbMATH Keywords: mean curvature
MaRDI profile type: Publication
full work available at URL: s00574-004-0009-8
OpenAlex ID: W2071687083
DOI: 10.1007/S00574-004-0009-8
description: scientific article; zbMATH DE number 2133667