Stability of non-singular group orbits
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 \(n<m\).
scientific article; zbMATH DE number 168268