论文标题
Lorentzian合成空间中的最佳运输，合成时间式的RICCI曲率下限和应用
Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications
论文作者
论文摘要
本工作的目标是三倍。 第一个目标是设定洛伦兹（前）长度空间中最佳运输的基础结果，包括周期性的单调性，最佳耦合的稳定性和坎托洛维奇二元性（对于光滑的洛伦兹多种歧管而言，有几个结果是新的）。 第二个是使用最佳传输的``lorentzian预先长度''的``序列式曲率''的合成概念。关键思想是分析熵函数的凸性特性，沿未来的定向概率测量学概率测量方案。事实证明，在测得的洛伦兹预性空间的合适弱收敛下，该概念是稳定的，从而瞥见了我们提出的方法的强度。 第三个目标是绘制应用程序，最著名的是扩展体积比较，并将霍金奇异定理（以尖锐的形式）扩展到合成环境。 洛伦兹（Lorentzian）预性空间的框架包括一个显着的示例类别：空间时间赋予了因果关系普通（或更强烈的局部Lipschitz）连续的Lorentzian指标，封闭的圆锥结构，一些量子重力的方法。
The goal of the present work is three-fold. The first goal is to set foundational results on optimal transport in Lorentzian (pre-)length spaces, including cyclical monotonicity, stability of optimal couplings and Kantorovich duality (several results are new even for smooth Lorentzian manifolds). The second one is to give a synthetic notion of ``timelike Ricci curvature bounded below and dimension bounded above'' for a measured Lorentzian pre-length space using optimal transport. The key idea being to analyse convexity properties of Entropy functionals along future directed timelike geodesics of probability measures. This notion is proved to be stable under a suitable weak convergence of measured Lorentzian pre-length spaces, giving a glimpse on the strength of the approach we propose. The third goal is to draw applications, most notably extending volume comparisons and Hawking singularity Theorem (in sharp form) to the synthetic setting. The framework of Lorentzian pre-length spaces includes as remarkable classes of examples: space-times endowed with a causally plain (or, more strongly, locally Lipschitz) continuous Lorentzian metric, closed cone structures, some approaches to quantum gravity.
