Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space

TL;DR

This paper investigates the stability of the volume preserving mean curvature flow in hyperbolic space, showing that under certain conditions, the flow exists indefinitely and converges exponentially to a sphere.

Contribution

It extends Huisken's Euclidean results to hyperbolic space, proving exponential convergence for hypersurfaces close to spheres in the hyperbolic setting.

Findings

Flow exists for all time under hyperbolic mean convexity.

Flow converges exponentially to an umbilical sphere.

Stability results are established for initial hypersurfaces close to spheres.

Abstract

We consider the dynamic property of the volume preserving mean curvature flow. This flow was introduced by Huisken who also proved it converges to a round sphere of the same enclosed volume if the initial hypersurface is strictly convex in Euclidean space. We study the stability of this flow in hyperbolic space. In particular, we prove that if the initial hypersurface is hyperbolically mean convex and close to an umbilical sphere in the $L^2$-sense, then the flow exists for all time and converges exponentially to an umbilical sphere.