Theory of optimal transport for Lorentzian cost functions

零超曲面上的最优传输与零能量条件

Optimal transport on null hypersurfaces and the null energy condition

本文研究 Lorentzian 流形零超曲面上的退化最优传输问题，利用概率测度沿零测地线的 Boltzmann--Shannon 熵凸性刻画零能量条件，并推导出时空稳定性、零锥比较及 Hawking 面积定理等锐利应用。

通过最优输运视角看零能量条件：光滑与非光滑时空

The Null Energy Condition through the Lens of Optimal Transport: Smooth and Non-Smooth Spacetimes

本文综述了通过最优输运刻画零能量条件的研究，将其推广至非光滑时空，并建立了合成曲率条件及相应的奇点定理。

左不变三维接触子洛伦兹结构中最长弧的存在性

Existence of the longest arcs for left-invariant three-dimensional contact sub-Lorentzian structures

本文研究了左不变三维接触子洛伦兹结构中最长弧的存在性，提出了在可解李群和SL(2, R)李群 universal cover 上的充分存在条件。

全局双曲时空上弱 Kantorovich 势的最优输运与正则性

Optimal transport and regularity of weak Kantorovich potentials on a globally hyperbolic spacetime

本文研究全局双曲时空上代价为 $c_2$ 的最优输运问题，通过分析弱 Kantorovich 势（$\pi$-解）的正则性，证明两 Borel 概率测度间最优输运映射的存在性、唯一性与结构。

洛伦兹Lasry-Lions正则化定理

A Lorentzian Lasry-Lions regularization theorem

本文建立了一个一般的洛伦兹Lasry-Lions正则化定理，通过前向Lax-Oleinik演化与局部半凹性条件，得到函数的C^{1,1}_{loc}正则性，并应用于最优传输中的对偶最大化对。

因果集的Ollivier-Ricci曲率

Ollivier-Ricci curvature for causal sets

利用洛伦兹最优传输引入因果集的Ollivier-Ricci曲率，通过离散化方法恢复时间型Ricci曲率，并证明局部到全局传播结果和类时Bonnet-Myers定理。

基于最优输运的低正则性时空的类时Ricci界

Timelike Ricci bounds for low regularity spacetimes by optimal transport

该研究针对低正则性时空，利用最优输运方法证明了分布意义下类时Ricci有下界的C¹正则整体双曲时空满足类时测度收缩性质及相关几何不等式，C¹,¹正则下还成立更强曲率-维数条件且耦合与测地线唯一。

Regularity for the optimal transportation problem with Euclidean distance squared cost on the embedded sphere

We give sufficient conditions on initial and target measures supported on the sphere $§^n$ to ensure the solution to the optimal transport problem with the cost $|x-y|^2/2$ is a diffeomorphism.

Parabolic optimal transport equations on manifolds

We study a parabolic equation for finding solutions to the optimal transport problem on compact Riemannian manifolds with general cost functions. We show that if the cost satisfies the strong MTW condition and the stay-away singularity property, then the solution to the parabolic flow with any appropriate initial condition exists for all time and it converges exponentially to the solution to the optimal transportation problem. Such results hold in particular, on the sphere for the distance squared cost of the round metric and for the far-field reflector antenna cost, among others.