  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.
