Existence and Uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds

TL;DR

This paper proves the existence and uniqueness of stable constant mean curvature foliations in asymptotically hyperbolic 3-manifolds resembling Anti-de Sitter-Schwarzschild metrics, relevant in Einstein's equations with negative cosmological constant.

Contribution

It establishes the first rigorous proof of such foliations' existence and uniqueness in this geometric setting, extending previous work on asymptotically flat manifolds.

Findings

Existence of stable CMC foliations in asymptotically hyperbolic 3-manifolds.

Uniqueness of these foliations under specified conditions.

Relevance to solutions of Einstein's equations with negative cosmological constant.

Abstract

We prove existence and uniqueness of foliations by stable spheres with constant mean curvature for 3-manifolds which are asymptotic to Anti-de Sitter-Schwarzschild metrics with positive mass. These metrics arise naturally as spacelike timeslices for solutions of the Einstein equation with a negative cosmological constant.