A new optimal estimate for the first stability eigenvalue of closed hypersurfaces in Riemannian space forms
type: Item
instance of: scholarly article
title: A new optimal estimate for the first stability eigenvalue of closed hypersurfaces in Riemannian space forms
zbMATH Open document ID: 1409.53054
publication date: 2018-12-12T00:00:00Z
review text: Suppose that \(\Sigma^n\) is a closed \(n\)-dimensional hypersurface of constant mean curvature \(H\) in a complete, simply connected \((n+1)\)-dimensional space form \(Q_c^{n+1}\) of constant curvature \(c \in \{ -1, 0, 1\}\). The stability operator \(J\), which acts on smooth functions on \(\Sigma\), is defined by the formula \[ J = \Delta + |\Phi| + c(H^2+c), \] where \(\Delta\) is the Laplace operator on \(\Sigma\) with its induced metric, and \(|\Phi|\) is the norm of the trace-free part \(\Phi\) of the shape operator \(A\) of \(\Sigma\) in \(Q_c\). The operator \(\Phi\) vanishes at the umbilic points of \(\Sigma\). Under the additional hypothesis that \(\Phi\) satisfies a certain Okumura-type inequality, which is automatically satisfied whenever \(\Sigma\) has exactly two principal curvatures of constant multiplicity, the authors find an upper bound on the first eigenvalue of \(J\) in terms of \(H\), \(c\), \(n\) and \(\Phi\). This upper bound is attained for Clifford tori in the sphere \(S^{n+1}\), but it is not attained for any \(\Sigma\) in Euclidean (\(c=0\)) or hyperbolic (\(c=-1\)) space. The examples of hyperbolic cylinders in hyperbolic space show that the upper bound is not valid in general when \(\Sigma\) is no longer assumed to be compact.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 6992787
zbMATH Keywords: Riemannian space forms
zbMATH Keywords: closed \(H\)-hypersurfaces
zbMATH Keywords: strong stability
zbMATH Keywords: first stability eigenvalue
zbMATH Keywords: constant mean curvature
zbMATH Keywords: Clifford torus, circular and hyperbolic cylinders
zbMATH Keywords: totally umbilical hypersurface
Wikidata QID: Q115376998
author: Eudes L. De Lima
author: Henrique Fernandes de Lima
reviewed by: James Hebda
MaRDI profile type: Publication
full work available at URL: s12215-018-0332-3
OpenAlex ID: W2789741342
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DOI: 10.1007/S12215-018-0332-3
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 6992787