Volume preserving mean curvature flow of round surfaces in asymptotically flat spaces

TL;DR

This paper proves that under certain conditions, the volume preserving mean curvature flow of round surfaces in asymptotically flat 3-manifolds exists indefinitely and converges to a stable constant mean curvature surface, extending classical results in the field.

Contribution

It extends Huisken-Yau's classical result to asymptotically flat spaces and introduces an alternative method for constructing CMC foliations in such manifolds.

Findings

Flow exists for all time under positive ADM mass

Flow converges smoothly to a stable CMC surface

Provides an alternative approach to CMC foliation construction

Abstract

We study the volume preserving mean curvature flow of a surface immersed in an asymptotically flat $3$-manifold modeling an isolated gravitating system in General Relativity. We show that, if the ambient manifold has positive ADM mass and the initial surface is round in a suitable sense, then the flow exists for all times and converges smoothly to a stable CMC surface. This extends to the asymptotically flat setting a classical result by Huisken-Yau (Invent. Math. 1996) and allows to construct a CMC foliation of the outer part of the manifold by an alternative approach to the ones by Nerz (Calc. Var. PDE, 2015) or by Eichmair-Koerber (J. Diff. Geometry, 2024).