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 nonnegative
mean curvature and decaying volume.
Keywords
Lorentzian manifold, timelike incompleteness, CMC
foliation, general relativity
