On the CMC foliation of future ends of a spacetime

Abstract

We consider spacetimes with compact Cauchy hypersurfaces and with Ricci tensor bounded from below on the set of timelike unit vectors, and prove that the results known for spacetimes satisfying the timelike convergence condition, namely, foliation by CMC hypersurfaces, are also valid in the present situation, if corresponding further assumptions are satisfied. In addition we show that the volume of any sequence of spacelike hypersurfaces, which run into the future singularity, decays to zero provided there exists a time function covering a future end, such that the level hypersurfaces have non-negative mean curvature and decaying volume.