Abstract
We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along timelike curves, and an upper bound on the mean curvature of the hypersurface. Using these results, we give a new proof of Hawking’s singularity theorem.
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Treude, JH., Grant, J.D.E. Volume comparison for hypersurfaces in Lorentzian manifolds and singularity theorems. Ann Glob Anal Geom 43, 233–251 (2013). https://doi.org/10.1007/s10455-012-9343-z
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DOI: https://doi.org/10.1007/s10455-012-9343-z