Stability of volume and area preserving mean curvature flow in asymptotic Schwarzschild space

TL;DR

This paper proves that volume and area preserving mean curvature flows in asymptotic Schwarzschild space, starting near a sphere, exist globally and converge exponentially to a constant mean curvature sphere, establishing existence of such hypersurfaces.

Contribution

It extends stability results of mean curvature flows to asymptotically Schwarzschild spaces, including cases with pinched curvature, using center manifold analysis.

Findings

Flows exist globally for initial hypersurfaces close to spheres.

Flows converge exponentially to CMC hypersurfaces.

Establishes existence of CMC hypersurfaces in asymptotically flat space.

Abstract

In this paper, we investigate the stability of the volume preserving mean curvature flow (VPMCF) and area preserving mean curvature flow (APMCF) in the Schwarzschild space. We show that if the initial hypersurface is sufficiently close to a coordinate sphere, these flows exist globally and converge smoothly to a constant mean curvature (CMC) hypersurface, namely a coordinate sphere. For asymptotically Schwarzschild space, if the initial hypersurface has pinched curvature outside of some large compact set, or more orecisely sufficiently close to an isoperimetric hypersurface, outside of some large compact set in C^2 sense, we will apply similar method combined with the center manifold analysis to see that the flow still exists for all time and converges to CMC hypersurface exponentially fast. This in particular gives an existence result for a CMC hypersurface in asymptotically flat space.