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.
"Isoperimetric and Weingarten surfaces in the Schwarzschild manifold." J. Differential Geom. 94 (3) 387 - 407, July 2013. https://doi.org/10.4310/jdg/1370979333