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Volume preserving spacetime mean curvature flow and foliations of initial data sets

Jacopo Tenan

TL;DR

The work extends Huisken–Yau style volume-preserving curvature flow to a spacetime setting by introducing a generalized speed $\mathcal{H}=\sqrt[q]{H^q-|P|^q}$ on $C_{\frac12+\delta}^2$-asymptotically flat initial data with positive ADM energy. By combining a spectral analysis of the stability operator with a carefully chosen roundness class, the authors prove global existence and exponential convergence of the flow to a surface with constant spacetime curvature, thereby producing a CSTMC foliation that serves as a robust center-of-mass construction in General Relativity. The results recover the Cederbaum–Sakovich CSTMC foliation when $q=2$ and provide a broader framework for understanding how extrinsic curvature flows behave in asymptotically flat spacetimes, with implications for defining conserved quantities in isolated systems. Overall, the paper broadens the analytic toolkit for geometric flows in GR and strengthens the link between foliations by spacetime curvature and notions of center of mass in gravitating systems.

Abstract

We consider a volume preserving curvature evolution of surfaces in an asymptotically Euclidean initial data set with positive ADM-energy. The speed is given by a nonlinear function of the mean curvature which generalizes the spacetime mean curvature recently considered by Cederbaum-Sakovich (Calc. Var. PDE, 2021). Following a classical approach by Huisken-Yau (Invent. Math., 1996), we show that the flow starting from suitably round initial surfaces exists for all times and converges to a constant (spacetime) curvature limit. This provides an alternative construction of the CSTMC foliation by Cederbaum-Sakovich and has applications in the definition of center of mass of an isolated system in General Relativity.

Volume preserving spacetime mean curvature flow and foliations of initial data sets

TL;DR

The work extends Huisken–Yau style volume-preserving curvature flow to a spacetime setting by introducing a generalized speed on -asymptotically flat initial data with positive ADM energy. By combining a spectral analysis of the stability operator with a carefully chosen roundness class, the authors prove global existence and exponential convergence of the flow to a surface with constant spacetime curvature, thereby producing a CSTMC foliation that serves as a robust center-of-mass construction in General Relativity. The results recover the Cederbaum–Sakovich CSTMC foliation when and provide a broader framework for understanding how extrinsic curvature flows behave in asymptotically flat spacetimes, with implications for defining conserved quantities in isolated systems. Overall, the paper broadens the analytic toolkit for geometric flows in GR and strengthens the link between foliations by spacetime curvature and notions of center of mass in gravitating systems.

Abstract

We consider a volume preserving curvature evolution of surfaces in an asymptotically Euclidean initial data set with positive ADM-energy. The speed is given by a nonlinear function of the mean curvature which generalizes the spacetime mean curvature recently considered by Cederbaum-Sakovich (Calc. Var. PDE, 2021). Following a classical approach by Huisken-Yau (Invent. Math., 1996), we show that the flow starting from suitably round initial surfaces exists for all times and converges to a constant (spacetime) curvature limit. This provides an alternative construction of the CSTMC foliation by Cederbaum-Sakovich and has applications in the definition of center of mass of an isolated system in General Relativity.
Paper Structure (12 sections, 23 theorems, 172 equations)

This paper contains 12 sections, 23 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Definitions and basic properties
  4. Surfaces in initial data sets
  5. Round surfaces
  6. Spectral theory
  7. Mass and stability operator
  8. Volume preserving spacetime mean curvature flow
  9. Definition of the flow and evolution equations
  10. Evolution of integral quantities
  11. Evolution of $\|\mathcal{H}-\hbar\|_{L^2(\Sigma_t)}$ and convergence
  12. CSTMC foliation and centers of mass

Key Result

Theorem 1.3

Let $(\textup{M},\overline\textup{g},\overline\textup{K})$ be a constrained $C_{\frac{1}{2}+\delta}^2$-asymptotically flat initial data set in the sense of Definition def1 and def2, and suppose that $E_\textup{ADM}>0$. Fix $q\geq 2$. Let $\iota:\Sigma\hookrightarrow \textup{M}$ be a closed CMC-surfa where $r_\Sigma:=\min_{x\in\Sigma}|\vec{x}(x)|$. Then, there exists $\sigma_0=\sigma_0(C_0,\overlin

Theorems & Definitions (55)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Remark 2.6
  • ...and 45 more