We show that any star-shaped convex hypersurface with constantWeingarten curvature in the deSitter-Schwarzschildmanifold
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.