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

TL;DR

This paper introduces a volume-preserving curvature flow in asymptotically Euclidean initial data sets, demonstrating long-term existence and convergence to a constant spacetime curvature, aiding in defining the system's center of mass in General Relativity.

Contribution

It extends the spacetime mean curvature flow to a volume-preserving setting and provides a new construction of CSTMC foliations with applications in gravitational physics.

Findings

Flow exists for all time from suitable initial surfaces.

Flow converges to a constant spacetime curvature.

Application to defining center of mass in General Relativity.

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.