Foliation by Constant Mean Curvature Spheres on Asymptotically Flat Manifolds

TL;DR

This paper proves the existence and uniqueness of foliations by constant mean curvature spheres on asymptotically flat manifolds with nonzero ADM mass across all dimensions, extending prior results in the positive mass case.

Contribution

It establishes the existence and uniqueness of such foliations on a broader class of manifolds, including all dimensions and nonzero ADM mass.

Findings

Foliations exist uniquely in all dimensions for manifolds with nonzero ADM mass.

The results extend previous work limited to positive mass cases.

Provides a comprehensive framework for understanding geometric structures at infinity.

Abstract

In this paper, the existence and uniqueness of foliations by constant mean curvature spheres on asymptotically flat manifolds of nonzero ADM mass in all dimensions were established. (A similar result in the case of positive mass was obtained independently by G. Huisken and S. T. Yau, see the introduction of this paper and their paper in Inv. Math.)