Cosmological time versus CMC time in spacetimes of constant curvature

Abstract

In this paper, we investigate the existence of foliations by constant mean curvature (CMC) hypersurfaces in maximal, globally hyperbolic, spatially compact, spacetimes of constant curvature.

In the non-positive curvature case (i.e. for flat and locally anti-de Sitter spacetimes), we prove the existence of a global foliation of the spacetime by CMC Cauchy hypersurfaces. The positive curvature case (i.e. locally de Sitter spacetimes) is more delicate: in general, we are only able to prove the existence of a foliation by CMC Cauchy hypersurfaces in a neighbourhood of the past (or future) singularity.

Except in some exceptional and elementary cases, the leaves of the foliation we construct are the level sets of a time function, and the mean curvature of the leaves increases with time. In this case, we say that the spacetime admits a CMC time function.

Our proof is based on using the level sets of the cosmological time function as barriers. A major part of the work consists of proving the required curvature estimates for these level sets. One of the difficulties is the fact that the local behaviour of the cosmological time function near one point depends on the global geometry of the spacetime.