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

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