AI 中文总结
研究全局双曲时空固定点相关的聚焦时间和割线时间的性质，采用伊藤 - 田中的方法并适用于洛伦兹设置，证明了它们的局部利普希茨连续性，推导相关估计和准则，得出割迹的豪斯多夫余维数结论。
AI 中文摘要
我们证明，对于全局双曲时空中的一个固定点，聚焦时间在其有限的未来因果锥的开子集上是局部利普希茨连续的。我们还表明，割线时间在任何类时切向量的邻域内是局部利普希茨连续的，其相关测地线至少定义到（并包括）其割线时间。此外，我们推导了零锥附近利普希茨常数的定量估计，并提供了一个确保利普希茨性质扩展到零方向的准则。结果表明，一点的割迹具有至少为1的豪斯多夫余维数。这些结果扩展了伊藤 - 田中以及李 - 尼伦伯格关于完备黎曼流形的经典利普希茨连续性结果。我们的方法遵循伊藤 - 田中的方法，并适当地调整到洛伦兹设置。
英文摘要
We prove that, for a fixed point in a globally hyperbolic spacetime, the focalization time is locally Lipschitz continuous on the open subset of the future causal cone where it is finite. We also show that the cut time is locally Lipschitz continuous in a neighborhood of any timelike tangent vector whose associated geodesic is defined at least up to (and including) its cut time. Furthermore, we derive quantitative estimates for the Lipschitz constant near the null cone and provide a criterion ensuring that the Lipschitz property extends to null directions. As a consequence, we show that the cut locus of a point has Hausdorff codimension at least $1$. These results extend classical Lipschitz continuity results for complete Riemannian manifolds due to Itoh-Tanaka and Li-Nirenberg. Our approach follows the method of Itoh-Tanaka, suitably adapted to the Lorentzian setting.
