论文标题
Lorentzian度量空间及其Gromov-Hausdorff收敛
Lorentzian metric spaces and their Gromov-Hausdorff convergence
论文作者
论文摘要
我们提出了一种抽象的方法，用于洛伦兹Gromov-Hausdorff距离和收敛性，以及一种不使用辅助``正签名''指标或其他未观察到的lorentzian长度空间的替代方法。我们首先定义了（摘要）有界的洛伦兹 - 金属空间的概念，该空间足够通用，可以包含紧凑的因果凸出的全球空位和因果关系。我们定义了Gromov-Hausdorff的距离，并表明零GH距离的两个有界的Lorentzian-Metric空间确实是同量和同构的。然后，我们展示了如何从洛伦兹距离，拓扑，因果关系和这些空间的因果曲线来定义，从而获得有用的极限曲线定理。接下来，我们通过合适的（最大）圆相连特性来定义洛伦兹（长度）前长空间。这些定义在GH限制下被证明是稳定的。此外，我们为洛伦兹长度空间定义了截面曲率的边界，并证明它们在GH极限下也是稳定的。我们以（前）紧凑定理结束。
We present an abstract approach to Lorentzian Gromov-Hausdorff distance and convergence, and an alternative approach to Lorentzian length spaces that does not use auxiliary ``positive signature'' metrics or other unobserved fields. We begin by defining a notion of (abstract) bounded Lorentzian-metric space which is sufficiently general to comprise compact causally convex subsets of globally hyperbolic spacetimes and causets. We define the Gromov-Hausdorff distance and show that two bounded Lorentzian-metric spaces at zero GH distance are indeed both isometric and homeomorphic. Then we show how to define from the Lorentzian distance, beside topology, the causal relation and the causal curves for these spaces, obtaining useful limit curve theorems. Next, we define Lorentzian (length) prelength spaces via suitable (maximal) chronal connectedness properties. These definitions are proved to be stable under GH limits. Furthermore, we define bounds on sectional curvature for our Lorentzian length spaces and prove that they are also stable under GH limits. We conclude with a (pre)compactness theorem.
