Volume comparison for $C^{1,1}$ metrics

TL;DR

This paper extends classical geometric comparison theorems to Riemannian and Lorentzian manifolds with $C^{1,1}$-metrics, providing new proofs of Myers' and Hawking's singularity theorems under lower regularity assumptions.

Contribution

It develops volume comparison results and geometric inequalities for $C^{1,1}$-metrics, enabling the extension of key theorems to less smooth settings.

Findings

Established volume comparison for $C^{1,1}$-metrics

Proved Myers' theorem in $C^{1,1}$-regularity

Proved Hawking's singularity theorem in $C^{1,1}$-regularity

Abstract

We establish volume comparison results for balls in Riemannian manifolds with $C^{1,1}$-metrics with a lower bound on the Ricci tensor and for the evolution of spacelike, acausal, causally complete hypersurfaces with an upper bound on the mean curvature in spacetimes with $C^{1,1}$-metrics with a lower bound on the timelike Ricci curvature. These results are then used to give proofs of Myers' theorem and of Hawking's singularity theorem in this regularity.