Stability of the area preserving mean curvature flow in asymptotic Schwarzschild space

TL;DR

This paper proves long-time existence and exponential convergence of the area preserving mean curvature flow in asymptotic Schwarzschild spaces, providing new insights into foliation and uniqueness of constant mean curvature surfaces.

Contribution

It introduces a novel approach to establishing the existence and uniqueness of constant mean curvature surfaces in asymptotically Schwarzschild spaces.

Findings

Flow exists for all time under small traceless second fundamental form

Flow converges exponentially to a round sphere or constant mean curvature surface

Provides a new proof of foliation existence and uniqueness in this setting

Abstract

We first demonstrate that the area preserving mean curvature flow of hypersurfaces in space forms exists for all time and converges exponentially fast to a round sphere if the integral of the traceless second fundamental form is sufficiently small. Then we show that from sufficiently large initial coordinate sphere, the area preserving mean curvature flow exists for all time and converges exponentially fast to a constant mean curvature surface in 3-dimensional asymptotically Schwarzschild spaces. This provides a new approach to the existence of foliation established by Huisken and Yau. And also a uniqueness result follows