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 $$\mathcal {C}^{1,1}$$C1,1 (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 $$\mathcal {C}^{1,1}$$C1,1-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.

中文翻译：

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$$\mathcal {C}^{1,1}$$ C 1 , 1 -metrics 的体积比较

本文的目的是将用于平滑黎曼或洛伦兹流形的某些体积比较定理（Bishop-Gromov 和 Treude 和 Grant 的最新结果，Ann Global Anal Geom, 43:233–251, 2013）推广到仅为 $ $\mathcal {C}^{1,1}$$C1,1（可与 Lipschitz 连续导数微分）。特别地，我们建立（使用近似方法）一个体积单调性结果，用于空间状的、非因果的、未来因果完备的（即，任何过去的因果锥与超曲面的交集相对紧凑）超曲面的紧凑子集的演化。以 $$\mathcal {C}^{1,1}$$C1,1-metric 约束全局双曲时空的平均曲率，其下界为类时 Ricci 曲率，提供所有类时测地线set 存在的时间足够长。作为中间步骤，我们还展示了这种超曲面的切割轨迹在这个规律中仍然具有零度量——将众所周知的结果推广到平滑度量。为了表明这些体积比较结果有一些非常好的应用，我们随后给出了 Myers 定理、全局双曲时空的简单奇点定理和霍金奇点定理直接在这个规律中的证明。