Foliations by stable spheres with constant mean curvature for isolated systems without asymptotic symmetry

TL;DR

This paper extends the existence and uniqueness of constant mean curvature foliations in three-dimensional asymptotically flat manifolds, requiring only asymptotic flatness and positive mass, and relates the CMC-center of mass to the ADM-center.

Contribution

It generalizes prior results by weakening assumptions needed for CMC-foliations and characterizes the CMC-center of mass in broader settings.

Findings

Existence and uniqueness of CMC-foliation under minimal assumptions.

Identification of the CMC-center of mass with the ADM-center.

Extension of previous characterizations to less restrictive conditions.

Abstract

In 1996, Huisken-Yau showed that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed CMC-surfaces if it is asymptotically equal to the (spatial) Schwarzschild solution and has positive mass. Their assumptions were later weakened by Metzger, Huang, Eichmair-Metzger and others. We further generalize these existence results in dimension three by proving that it is sufficient to assume asymptotic flatness and non-vanishing mass to conclude the existence and uniqueness of the CMC-foliation and explain why this seems to be the conceptually optimal result. Furthermore, we generalize the characterization of the corresponding coordinate CMC-center of mass by the ADM-center of mass proven previously by Corvino-Wu, Huang, Eichmair-Metzger and others (under other assumptions).