Isoperimetric and Weingarten surfaces in the Schwarzschild manifold

TL;DR

This paper proves that star-shaped convex hypersurfaces with constant Weingarten curvature in the Schwarzschild manifold are spherical, and it characterizes isoperimetric surfaces in the doubled Schwarzschild manifold for various volumes.

Contribution

It establishes the uniqueness of spherical hypersurfaces with constant Weingarten curvature and fully describes large-volume isoperimetric surfaces in the Schwarzschild setting.

Findings

Star-shaped convex hypersurfaces with constant Weingarten curvature are spheres.

Existence of isoperimetric surfaces for all enclosed volumes.

Complete description of large-volume isoperimetric surfaces.

Abstract

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.