Rigidity and compactness with constant mean curvature in warped product manifolds

TL;DR

This paper establishes the rigidity of boundaries with constant mean curvature in certain warped product manifolds, including models relevant to General Relativity, and characterizes limits of such boundaries with converging mean curvatures.

Contribution

It proves a new distributional CMC-rigidity result for rectifiable boundaries in warped product manifolds, extending previous understanding to limits of boundaries with converging mean curvatures.

Findings

Rigidity of rectifiable boundaries with constant distributional mean curvature.

Characterization of limits of boundaries with converging mean curvatures.

Application to models in General Relativity like deSitter--Schwarzschild.

Abstract

We prove the rigidity of rectifiable boundaries with constant distributional mean curvature in the Brendle class of warped product manifolds (which includes important models in General Relativity, like the deSitter--Schwarzschild and Reissner--Nordstrom manifolds). As a corollary we characterize limits of rectifiable boundaries whose mean curvatures converge, as distributions, to a constant. The latter result is new, and requires the full strength of distributional CMC-rigidity, even when one considers smooth boundaries whose mean curvature oscillations vanish in arbitrarily strong $C^k$-norms.