  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.
