On the global evolution problem in 2+1 gravity

TL;DR

This paper proves the existence of global constant mean curvature foliations in certain 2+1 dimensional Lorentzian manifolds, linking geometric analysis with solutions to Einstein's equations.

Contribution

It establishes the existence of global CMC foliations in 2+1 gravity models with high genus surfaces, using Hamiltonian reduction and Teichmüller space techniques.

Findings

Existence of global CMC foliations in 2+1 gravity models.

Reduction of Einstein equations to a Hamiltonian system on Teichmüller space.

Key estimates of Dirichlet energy are used in the proof.

Abstract

Existence of global CMC foliations of constant curvature 3-dimensional maximal globally hyperbolic Lorentzian manifolds, containing a constant mean curvature hypersurface with $\genus(\Sigma) > 1$ is proved. Constant curvature 3-dimensional Lorentzian manifolds can be viewed as solutions to the 2+1 vacuum Einstein equations with a cosmological constant. The proof is based on the reduction of the corresponding Hamiltonian system in constant mean curvature gauge to a time dependent Hamiltonian system on the cotangent bundle of Teichm\"uller space. Estimates of the Dirichlet energy of the induced metric play an essential role in the proof.