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

Carlo Sinestrari, Jacopo Tenan

TL;DR

This work extends Huisken-Yau’s CMC-foliation result to general $C_{\frac{1}{2}+\delta}^2$-asymptotically flat 3-manifolds with positive ADM mass by evolving initial round surfaces under volume-preserving mean curvature flow. The authors combine a carefully defined class of round surfaces with spectral analysis of the stability operator to obtain global existence and smooth convergence to a stable CMC leaf, and they derive quantitative control of the barycenter and translational modes. Key tools include integral curvature estimates, De Lellis–Müller-type regularity, and Regge-Teitelboim symmetry, which together allow an alternative flow-based construction of the CMC foliation in the outer region and yield a dynamical description of stability. The results provide an input for canonical radial coordinates and center-of-mass definitions in mathematical relativity and connect to prior elliptic approaches by Nerz and Eichmair–Koerber, while highlighting the role of positive mass in preventing drift of the foliation.

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).

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

TL;DR

This work extends Huisken-Yau’s CMC-foliation result to general -asymptotically flat 3-manifolds with positive ADM mass by evolving initial round surfaces under volume-preserving mean curvature flow. The authors combine a carefully defined class of round surfaces with spectral analysis of the stability operator to obtain global existence and smooth convergence to a stable CMC leaf, and they derive quantitative control of the barycenter and translational modes. Key tools include integral curvature estimates, De Lellis–Müller-type regularity, and Regge-Teitelboim symmetry, which together allow an alternative flow-based construction of the CMC foliation in the outer region and yield a dynamical description of stability. The results provide an input for canonical radial coordinates and center-of-mass definitions in mathematical relativity and connect to prior elliptic approaches by Nerz and Eichmair–Koerber, while highlighting the role of positive mass in preventing drift of the foliation.

Abstract

We study the volume preserving mean curvature flow of a surface immersed in an asymptotically flat -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).
Paper Structure (11 sections, 25 theorems, 160 equations)

This paper contains 11 sections, 25 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions and basic properties
  4. Round surfaces
  5. Spectral theory
  6. Mass and the stability operator
  7. The translational part of the mean curvature
  8. Volume preserving mean curvature flow
  9. Definition of the flow and evolution equations
  10. Evolution of integral quantities
  11. Evolution of the barycenter and convergence

Key Result

Theorem 1.2

Let $(\textup{M},\overline\textup{g})$ a $C^4_2$-asymptotically Schwarzschild$3$-manifold, in the sense that asymptc8 is replaced by where $\overline \textup{g}^S$ is the Schwarzschild metric for some $m>0$ and $\partial^{|k|}$ denote derivatives of order $k$. Let $\Sigma_t=F(\Sigma,t)$ be the solution of the flow vpmcf with initial data given by the Euclidean coordinate sphere $\mathbb{S}_r(0)$,

Theorems & Definitions (58)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: $C_{1+\delta}^1$-Regge-Teitelboim conditions
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4: Round surfaces
  • Remark 2.5
  • Remark 2.6
  • Lemma 2.7
  • ...and 48 more