合成洛伦兹时空中的Monge输运问题
Monge's transport problem in synthetic Lorentzian spacetimes
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中文总结 AI 辅助
本文在满足前向类时测度收缩性质的合成洛伦兹时空上，利用针分解技术证明洛伦兹Monge问题解的存在性，并建立$TMCP^+(K,N)$与$TMCP^+_{rLip}(K,N)$的等价性。
中文摘要 AI 辅助
我们证明了在满足前向类时测度收缩性质$TMCP^+(K,N)$的合成洛伦兹时空上，洛伦兹Monge问题解的存在性。该问题的代价函数是时间分离函数$\ell(x,y)$（即洛伦兹距离），它表示粒子从$x$运动到$y$时所能经历的最大时间增量。该解基于针分解技术，将完整问题约化为一维对应问题。作为针分解的一个应用，我们证明了洛伦兹类时测度收缩性质$TMCP^+(K,N)$等价于$TMCP^+_{rLip}(K,N)$条件，其中曲率维数条件定义在反向1-Lipschitz函数的梯度流曲线上。
英文摘要
We prove the existence of solutions to the Lorentzian Monge problem on synthetic Lorentzian spacetimes that satisfy the forward timelike measure contraction property $TMCP^+(K,N)$. The cost for this problem is the time separation function $\ell(x,y)$ (the Lorentz distance), which represents the maximum amount a particle can age when traveling from $x$ to $y$. The solution is based on the needle decomposition technique, and amounts to a reduction of the full problem to its one-dimensional counterparts. As an application of needle decomposition, we show that the Lorentzian timelike measure contraction property $TMCP^+(K,N)$ is equivalent to the $TMCP^+_{rLip}(K,N)$ condition, where the curvature dimension conditions are defined on gradient flow curves of reverse 1-Lipschitz functions.
