论文标题
最佳运输的及时速度为低规律性空间的范围
Timelike Ricci bounds for low regularity spacetimes by optimal transport
论文作者
论文摘要
我们证明，具有$ \ smash {\ mathrm {c}^1} $的全球平滑时空，其ricci张量在所有时间表的方向上从下面的分布感中，以分配的态度从下面的角度来看，以分配的方式遵守时间表的测量属性。该结果包括一类具有边界规律性的空间，在该空间中，爱因斯坦方程的局部存在结果是在合成意义上具有及时的RICCI边界的空间中知道的。特别是，这些空位满足了序列的布鲁恩 - 山科斯基，帽子的美犬和主教 - 格罗莫夫主教的不平等现象，没有任何时间般的非分支假设。 如果该度量是$ \ smash {\ mathrm {c}^{1,1}} $，则实际上，较强的时间级曲率差异条件会保持。在这种规律性下，我们还获得了按时间顺序排列的最佳耦合和年代学测量学的唯一性。
We prove that a globally hyperbolic smooth spacetime endowed with a $\smash{\mathrm{C}^1}$-Lorentzian metric whose Ricci tensor is bounded from below in all timelike directions, in a distributional sense, obeys the timelike measure-contraction property. This result includes a class of spacetimes with borderline regularity for which local existence results for the vacuum Einstein equation are known in the setting of spaces with timelike Ricci bounds in a synthetic sense. In particular, these spacetimes satisfy timelike Brunn-Minkowski, Bonnet-Myers, and Bishop-Gromov inequalities in sharp form, without any timelike nonbranching assumption. If the metric is even $\smash{\mathrm{C}^{1,1}}$, in fact the stronger timelike curvature-dimension condition holds. In this regularity, we also obtain uniqueness of chronological optimal couplings and chronological geodesics.
