标题： 洛伦兹-格罗莫夫-豪斯多夫收敛性与预紧性 显示英文标题

标题： Lorentzian Gromov-Hausdorff convergence and pre-compactness

摘要： 本文的目标是引入一种类似于Gromov-Hausdorff收敛的洛伦兹空间收敛性概念，基于由因果钻石组成的$\epsilon$-网，并且仅依赖于时间间隔函数。 这产生了一个几何意义上的收敛性概念，可以应用于合成洛伦兹空间（洛伦兹预长度空间）或光滑时空。 在主要结果中，我们证明了著名的Gromov度量空间预紧致定理的一个洛伦兹版本，其中用球形覆盖被钻石形覆盖所取代。 这导致了全局双曲时空类别的几何预紧致性结果，这些时空满足柯西超曲面上的均匀加倍性质以及因果关系上的适当控制。 论文的最后一部分建立了几个应用：我们表明Chruściel-Grant近似是这里引入的洛伦兹Gromov-Hausdorff收敛的一个实例，我们证明了类时截面曲率界在这种收敛下是稳定的，我们引入了类时爆破切线并讨论了它与因果集理论主要猜想的联系。 显示英文摘要

摘要： The goal of the paper is to introduce a convergence \`a la Gromov-Hausdorff for Lorentzian spaces, building on $\epsilon$-nets consisting of causal diamonds and relying only on the time separation function. This yields a geometric notion of convergence, which can be applied to synthetic Lorentzian spaces (Lorentzian pre-length spaces) or smooth spacetimes. Among the main results, we prove a Lorentzian counterpart of the celebrated Gromov's pre-compactness theorem for metric spaces, where controlled covers by balls are replaced by controlled covers by diamonds. This yields a geometric pre-compactness result for classes of globally hyperbolic spacetimes, satisfying a uniform doubling property on Cauchy hypersurfaces and a suitable control on the causality. The final part of the paper establishes several applications: we show that Chru\'sciel-Grant approximations are an instance of the Lorentzian Gromov-Hausdorff convergence here introduced, we prove that timelike sectional curvature bounds are stable under such a convergence, we introduce timelike blow-up tangents and discuss connections with the main conjecture of causal set theory.