Foliation of constant harmonic mean curvature surfaces in asymptotic Schwarzschild spaces

TL;DR

This paper studies a volume-preserving curvature flow in asymptotic Schwarzschild spaces, showing it leads to a foliation of the space by constant harmonic mean curvature surfaces and relates to the ADM center of mass.

Contribution

It proves long-time existence, exponential convergence, and foliation properties of the flow, connecting geometric analysis with the ADM mass concept.

Findings

Flow converges to constant harmonic mean curvature surfaces

Surfaces form a foliation outside a large ball

Center of mass matches ADM definition

Abstract

This paper investigates the volume-preserving harmonic mean curvature flow in asymptotically Schwarzschild spaces. We demonstrate the long-time existence and exponential convergence of this flow with a coordinate sphere of large radius serving as the initial surface in the asymptotically flat end, which eventually converges to a constant harmonic mean curvature surface. We also establish that these surfaces form a foliation of the space outside a large ball. Finally, we utilize this foliation to define the center of mass, proving that it agrees with the center of mass defined by the ADM formulation of the initial data set.