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Second variation estimates for minimal orbits. (English) Zbl 0605.53025

Geometric measure theory and the calculus of variations, Proc. Summer Inst., Arcata/Calif. 1984, Proc. Symp. Pure Math. 44, 139-148 (1986).
[For the entire collection see Zbl 0577.00014.]
Let G be a compact Lie group of isometries of an m-dimensional Riemannian manifold M; assume that all the orbits of G are principal. Suppose that N is a minimal orbit of G in M, and set \(n=\dim (N)\); then N is a critical point of the volume function v, where v(p) denotes the volume of the orbit passing through p. Normalize the metric on M so \(v=1\) at the points of N. Let H be the Hessian of v at some point q of N, and let \(\lambda\) denote the least eigenvalue of H on the normal space to N at q. Then \(\lambda\geq 0\) is a necessary condition, but not a sufficient one, for N to be stable (i.e., to have nonnegative second variation).
As in his earlier paper [Trans. Am. Math. Soc. 294, 537-552 (1986; Zbl 0596.53050)] the author obtains sufficient conditions for stability by considering an orthonormal family of invariant vector fields \(W_ 1,...,W_{m-n}\) orthogonal to the orbits of G near N, and then letting \(W_{ik}\) denote the tangential component of \([W_ i,W_ k]\) along N. In the present paper the author also assumes that there is an invariant unit tangent field X on N such that (a) the integral curves of X are closed curves of length L, and (b) each \(W_{ik}\) is parallel to X. For each i, let \(v_ i\) be the number of k such that \(W_{ik}\neq 0\), and define \(c_{ik}\) by \(\sqrt{v_ iv_ k}W_{ik}=c_{ik}X.\)
Theorem: Each of the following statements implies that N is stable.
(A) \(\lambda \geq c^ 2_{ik}/4\) for all i, k.
(B) \(\lambda\geq 0\), and if i,k is any pair such that \(c^ 2_{ik}>4\lambda\), then \[ \frac{| c_{ik}| L}{2\pi}\leq \max \{1,\quad (\frac{0.8}{\mu_{ik}})(\frac{1-\mu_{ik}}{1- \mu_{ik}+2\mu^ 2_{ik}})\}, \] where \(\mu_{ik}=\sqrt{1-4c^{- 2}_{ik}\lambda}.\)
The author’s main tool is a formula for the second variation which he obtained in his earlier paper mentioned above. (In that paper he also proved the sufficiency condition (A), but without the extra hypotheses involving X.) The author also provides sufficient conditions to ensure the instability of N in the case X is a Killing field. (His method uses results of D. D. Bleecker [Trans. Am. Math. Soc. 275, 409-416 (1983; Zbl 0506.53021)] on the simultaneous eigenvalues of the Laplace operator \(\Delta\) on N and the operator \(-X^ 2.)\) In particular, he obtains an instability condition for the case G is the unitary group U(n), the orbit N is codimension 2 and equivariantly isometric to \(S^{2n-1}\), with U(n) acting in the usual way, and X corresponds to the usual \(S^ 1\) action.
Reviewer: R.Reilly

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
49Q05 Minimal surfaces and optimization
49Q20 Variational problems in a geometric measure-theoretic setting