Volume Comparison for Hypersurfaces in Lorentzian Manifolds and Singularity Theorems

TL;DR

This paper establishes comparison theorems for hypersurface volume evolution in Lorentzian manifolds with curvature bounds, leading to a new proof of Hawking's singularity theorem.

Contribution

It introduces novel volume comparison results for hypersurfaces in Lorentzian geometry and applies them to provide an alternative proof of Hawking's singularity theorem.

Findings

Established volume comparison theorems under curvature bounds

Provided a new proof of Hawking's singularity theorem

Enhanced understanding of hypersurface evolution in Lorentzian manifolds

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.