Existence and uniqueness of constant mean curvature foliations of general asymptotically hyperbolic 3-manifolds

TL;DR

This paper extends the existence and uniqueness of constant mean curvature foliations to a broader class of asymptotically hyperbolic 3-manifolds, linking these foliations to the concept of center of mass.

Contribution

It generalizes previous results by establishing CMC-foliations in manifolds with only a timelike mass vector, not requiring a mass aspect function.

Findings

Proves existence of CMC-foliation in wider class of manifolds.

Shows CMC-foliation characterizes the center of mass.

Extends previous theorems to more general asymptotically hyperbolic manifolds.

Abstract

In 1996, Huisen-Yau proved that every three-dimensional, asymptotically Schwarzschilden manifold with positive mass is uniquely foliated by stable spheres of constant mean curvature and they defined the center of mass using this CMC-foliation. Rigger and Neves-Tian showed in 2004 and 2009/10 analogous existence and uniqueness theorems for three-dimensional, asymptotically Anti-de Sitter and asymptotically hyperbolic manifolds with positive mass aspect function, respectively. Last year, Cederbaum-Cortier-Sakovich proved that the CMC-foliation characterizes the center of mass in the hyperbolic setting, too. In this article, the existence and the uniqueness of the CMC-foliation are further generalized to the wider class of asymptotically hyperbolic manifolds which do not necessarily have a well-defined mass aspect function, but only a timelike mass vector. Furthermore, we prove that the CMC-foliation also characterizes the center of mass in this more general setting.