Constant mean curvature foliations in cosmological spacetimes

TL;DR

This paper reviews conjectures and recent progress on the existence of constant mean curvature foliations in cosmological spacetimes, which are important for defining a preferred time in general relativity.

Contribution

It summarizes recent advances and methods in proving the existence of constant mean curvature foliations under certain conditions in cosmological spacetimes.

Findings

Progress on proving conjectures under additional assumptions

Explanation of proof methods used in recent results

Discussion of implications for cosmic censorship and universe recollapse

Abstract

Foliations by constant mean curvature hypersurfaces provide a possibility of defining a preferred time coordinate in general relativity. In the following various conjectures are made about the existence of foliations of this kind in spacetimes satisfying the strong energy condition and possessing compact Cauchy hypersurfaces. Recent progress on proving these conjectures under supplementary assumptions is reviewed. The method of proof used is explained and the prospects for generalizing it discussed. The relations of these questions to cosmic censorship and the closed universe recollapse conjecture are pointed out.