Stability of hypersurfaces of constant mean curvature in Riemannian manifolds
type: Item
instance of: scholarly article
title: Stability of hypersurfaces of constant mean curvature in Riemannian manifolds
zbMATH Open document ID: 0653.53045
DOI: 10.1007/BF01161634
author: J.-H. Eschenburg
author: J. Lucas M. Barbosa
author: Manfredo P. do Carmo
published in: Mathematische Zeitschrift
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/183722
review text: Let \(M_ 0\) be a compact oriented hypersurface of constant mean curvature \(H_ 0\) in an \((n+1)\)-dimensional Riemannian manifold \(\bar M.\) Such hypersurfaces arise as the stationary points of a constrained variational problem: minimize the (hypersurface) area functional A subject to fixed volume V. The method of Lagrange multipliers then realizes \(M_ 0\) as a stationary point for the unconstrained functional \(J=A+H_ 0V.\) The authors say that \(M_ 0\) is stable if the second variation \(\delta^ 2A\) is nonnegative for all volume-preserving deformations of \(M_ 0\); equivalently, if \(\delta^ 2J\geq 0\) for all such deformations. (In a review of several related papers [MR 85k:58021] R. Osserman refers to the latter formulation as `weak stability'.) The paper has three main results: (A): Suppose that \(\bar M\) is a simply- connected space form of curvature c. Then \(M_ 0\) is stable if and only if it is a geodesic sphere in M. (The case \(c=0\) was actually done in an earlier paper of the first two authors [ibid. 185, 339-353 (1984; Zbl 0513.53002)].) (B): Suppose that \(\bar M\) is a projective space, with the usual metric, over one of the fields \({\mathbb{R}}\), \({\mathbb{C}}\) or \({\mathbb{H}}\). If \(M_ 0\) is the boundary of a geodesic tube of radius \(\rho\) over a totally geodesic subspace of \(\bar M\) then \(M_ 0\) is stable if and only if \(\rho\) is sufficiently constricted (the paper provides precise bounds). In particular, there exist geodesic hyperspheres which are not stable, and there exist stable hypersurfaces which are not geodesic hyperspheres. (C): Suppose that \(\bar M\) is the complex hyperbolic space (i.e., the noncompact dual of the complex projective space). Then every geodesic hypersphere is stable. \{Reviewer's Comments: (a) The authors do not make clear why they prefer the formulation in terms of the functional J to the more natural formulation in terms of A. The usual reason for introducing Lagrange multipliers is to eliminate t…
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 4065868
zbMATH Keywords: constant mean curvature
zbMATH Keywords: volume-preserving deformations
zbMATH Keywords: stability
zbMATH Keywords: geodesic tube
zbMATH Keywords: second variation
Wikidata QID: Q115393983
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4065868