Foliations of asymptotically flat 3-manifolds by stable constant mean curvature spheres

TL;DR

This paper proves the existence of an asymptotic foliation by stable constant mean curvature spheres in asymptotically flat 3-manifolds with positive mass, and relates the geometric and Hamiltonian centers of mass, extending previous results with new methods.

Contribution

It provides a new proof of the foliation existence using Lyapunov-Schmidt reduction and characterizes large stable CMC spheres in 3D under nonnegative scalar curvature.

Findings

Existence of asymptotic foliation by stable CMC spheres in asymptotically flat manifolds.

Alignment of geometric and Hamiltonian centers of mass.

Uniqueness of large stable CMC spheres enclosing the center in 3D with nonnegative scalar curvature.

Abstract

Let $(M,g)$ be an asymptotically flat Riemannian manifold of dimension $n\geq 3$ with positive mass. We give a short proof based on Lyapunov-Schmidt reduction of the existence of an asymptotic foliation of $(M, g)$ by stable constant mean curvature spheres. Moreover, we show that the geometric center of mass of the foliation agrees with the Hamiltonian center of mass of $(M,g)$. In dimension $n = 3$, these results were shown previously by C. Nerz using a different approach. In the case where $n=3$ and the scalar curvature of $(M, g)$ is nonnegative, we prove that the leaves of the asymptotic foliation are the only large stable constant mean curvature spheres that enclose the center of $(M, g)$. This was shown previously under more restrictive decay assumptions and using a different method by S. Ma.