  We show that any star-shaped convex hypersurface with constant Weingarten
curvature in the deSitter-Schwarzschild manifold is a sphere of symmetry.
Moreover, we study an isoperimetric problem for bounded domains in the doubled
Schwarzschild manifold. We prove the existence of an isoperimetric surface for
any value of the enclosed volume, and we completely describe the isoperimetric
surfaces for very large enclosed volume. This complements work in H. Bray's
thesis, where isoperimetric surfaces homologous to the horizon are studied.
