On long-time evolution in general relativity and geometrization of 3-manifolds

TL;DR

This paper explores the connection between the long-term behavior of vacuum Einstein equations and the geometrization of 3-manifolds, focusing on CMC hypersurfaces and singularity avoidance.

Contribution

It establishes relations between Einstein evolution and 3-manifold geometrization, and discusses open problems linking these mathematical areas.

Findings

Relations between Einstein evolution and 3-manifold geometrization

Results on singularity avoidance of CMC foliations

Open problems in the field

Abstract

We describe some relations between the long-time asymptotic behavior of the vacuum Einstein evolution equations and the geometrization of 3-manifolds. These relations are expressed in terms of evolution of CMC hypersurfaces in the vacuum space-time.Some results are also obtained on the singularity avoidance of CMC foliations. In addition, the paper describes a number of open problems relating these two areas.