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

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.

23 pages, minor revisions, final version

References in corpus (4)

• Singularity Theorems and Their Consequences
• Limit curve theorems in Lorentzian geometry
• Global hyperbolicity for spacetimes with continuous metrics
• The Penrose singularity theorem in regularity $C^{1,1}$

Cited by in corpus (5)

• The Hawking-Penrose singularity theorem for $C^{1,1}$-Lorentzian metrics
• A review of Lorentzian synthetic theory of timelike Ricci curvature bounds
• Local Flexibility for Open Partial Differential Relations
• Volume Singularities in General Relativity
• Hawking-type singularity theorems for worldvolume energy inequalities