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.
arXiv Mathematics e-prints
- Pub Date:
- August 2004
- Mathematics - Differential Geometry;
- General Relativity and Quantum Cosmology;
- High Energy Physics - Theory
- 12 pages, a pdf version can also be retrieved from http://www.math.uni-heidelberg.de/studinfo/gerhardt/foliation2.pdf and bibtex data from http://www.math.uni-heidelberg.de/studinfo/gerhardt/bibtexcgfoliation2.html, v2: the result of Lemma 2.1 improved, Theorem 0.3 removed, because it is already known, v3: Minor changes, section numbering starts with 1