Stability of constant mean curvature surfaces in three dimensional warped product manifolds

TL;DR

This paper proves that stable, compact, nonzero constant mean curvature surfaces in certain warped product manifolds are topological spheres or slices, under conditions on the mean curvature and ambient space curvature.

Contribution

It generalizes stability and classification results for constant mean curvature surfaces to a broad class of warped product manifolds with curvature bounds.

Findings

Stable CMC surfaces are topological spheres under curvature bounds.

In de Sitter-Schwarzschild and Reissner-Nordstrom manifolds, stable CMC surfaces are slices.

Results apply to general Riemannian manifolds with curvature-dependent mean curvature bounds.

Abstract

In this paper we prove that stable, compact without boundary, oriented, nonzero constant mean curvature surfaces in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds are the slices, provided its mean curvature satisfies some positive lower bound. More generally, we prove that stable, compact without boundary, oriented nonzero constant mean curvature surfaces in a large class of three dimensional warped product manifolds are embedded topological spheres, provided the mean curvature satisfies a positive lower bound depending only on the ambient curvatures. We conclude the paper proving that a stable, compact without boundary, nonzero constant mean curvature surface in a general Riemannian is a topological sphere provided its mean curvature has a lower bound depending only on the scalar curvature of the ambient space and the squared norm of the mean curvature vector field of the immersion of the ambient space in some Euclidean space.