Volume comparison for C 1 , 1 -metrics

The aim of this paper is to generalize certain volume comparison theorems (Bishop-Gromov and a recent result of Treude and Grant, Ann Global Anal Geom, 43:233-251, 2013) for smooth Riemannian or Lorentzian manifolds to metrics that are only $C^{1,1}$ (differentiable with Lipschitz continuous derivatives). In particular we establish (using approximation methods) a volume monotonicity result for the evolution of a compact subset of a spacelike, acausal, future causally complete (i.e., the intersection of any past causal cone with the hypersurface is relatively compact) hypersurface with an upper bound on the mean curvature in a globally hyperbolic spacetime with a $C^{1,1}$ -metric with a lower bound on the timelike Ricci curvature, provided all timelike geodesics starting in this compact set exist long enough. As an intermediate step, we also show that the cut locus of such a hypersurface still has measure zero in this regularity-generalizing the well-known result for smooth metrics. To show that these volume comparison results have some very nice applications, we then give a proof of Myers' theorem, of a simple singularity theorem for globally hyperbolic spacetimes, and of Hawking's singularity theorem directly in this regularity.