Stability of non-singular group orbits
type: Item
instance of: scholarly article
title: Stability of non-singular group orbits
zbMATH Open document ID: 0791.53057
author: Clark D. Horton
published in: Pacific Journal of Mathematics
publication date: 1993-05-16T00:00:00Z
review text: Let \(G\) be a compact connected Lie group of isometries acting effective on a Riemannian manifold \(M\). Using \textit{J. E. Brothers'} second variation formula [Trans. Am. Math. Soc. 294, 537-552 (1986; Zbl 0596.53050)], the author establishes sufficient conditions for the stability of dimension one and codimension two minimal principal orbits. These conditions are expressed in terms of the eigenvalues of the Laplacian on the orbit, the length and eigenvalues of a \(G\)-invariant vector field and the eigenvalues of the Hessian matrix of the volume function \(\widetilde{\nu}: M\to\mathbb{R}\), \(\widetilde{\nu}(p)=\) the volume of the orbit through \(p\). In particular, the stability of equivariant minimal embeddings of codimension two spheres is derived. Sufficient conditions in order that an isolated exceptional orbit be stable are also given. These results are used to produce new examples of stable minimal submanifolds in the lens spaces \(L(p; q_ 1,\dots, q_ m)\) and in the quaternionic space-forms. It also follows that the standard inclusion of \(\mathbb{R} P^ n\) in \(\mathbb{R} P^ m\) is stable for \(n<m\).
reviewed by: Mircea Craioveanu
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 168268
zbMATH Keywords: minimal principal orbits
zbMATH Keywords: eigenvalues
zbMATH Keywords: Laplacian
zbMATH Keywords: \(G\)-invariant vector field
zbMATH Keywords: volume function
zbMATH Keywords: equivariant minimal embeddings
zbMATH Keywords: lens spaces
zbMATH Keywords: quaternionic space-forms
MaRDI profile type: Publication
full work available at URL: pjm.1992.156.135
OpenAlex ID: W2068144676
DOI: 10.2140/PJM.1992.156.135
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description: scientific article; zbMATH DE number 168268