Stability of hypersurfaces of constant mean curvature in Riemannian manifolds.(English)Zbl 0653.53045

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 the constraints on the allowed deformations; but this does not carry through to the second variation, and indeed the authors impose the same constraints on J as they would on A: volume preserving deformations. (b) E. Heinze [Math. Ann. 280, No.3, 389- 402 (1988; Zbl 0628.53044)] proves Theorem (A) in a different way.}
Reviewer: R.C.Reilly

MSC:

 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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References:

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