Continuing recent efforts in extending the classical singularity theorems of General Relativity to low regularity metrics, we give a complete proof of both the Hawking and the Penrose singularity theorem for $C^1$-Lorentzian metrics - a regularity where one still has existence but not uniqueness for solutions of the geodesic equation. The proofs make use of careful estimates of the curvature of approximating smooth metrics and certain stability properties of long existence times for causal geodesics. On the way we also prove that for globally hyperbolic spacetimes with a $C^1$-metric causal geodesic completeness is $C^1$-fine stable. This improves a similar older stability result of Beem and Ehrlich where they also used the $C^1$-fine topology to measure closeness but still required smoothness of all metrics. Lastly, we include a brief appendix where we use some of the same techniques in the Riemannian case to give a proof of the classical Myers Theorem for $C^1$-metrics.

中文翻译：

---

$$C^1$$-Lorentzian 度量的奇点定理

继续最近将广义相对论的经典奇点定理扩展到低正则性度量的努力，我们对 $C^1$-Lorentzian 度量的霍金和彭罗斯奇点定理给出了完整的证明 - 一个仍然存在但不存在的正则性测地线方程解的唯一性。这些证明利用了对近似平滑度量的曲率和因果测地线的长存在时间的某些稳定性属性的仔细估计。在此过程中，我们还证明对于具有 $C^1$-metric 因果测地线完整性的全局双曲时空是 $C^1$-fine 稳定的。这改进了 Beem 和 Ehrlich 类似的旧稳定性结果，他们也使用 $C^1$-fine 拓扑来测量接近度，但仍然需要所有指标的平滑度。最后，