论文标题
合成与分布较低的RICCI曲率边界
Synthetic versus distributional lower Ricci curvature bounds
论文作者
论文摘要
我们将两种标准方法比较，以定义较低的RICCI曲率界限，以低于$ c^2 $的规律性指标。一方面，这些是通过在最佳传输框架中熵函数的弱位移凸的合成定义，而分布的分布基于schwartz意义上的Ricci Tensor的分布性定义。事实证明，分布界意味着$ c^1 $的指标的熵范围，并且在额外的汇聚条件下，对$ c^{1,1} $的匡威（Converse）保留在公制的正规化条件下。
We compare two standard approaches to defining lower Ricci curvature bounds for Riemannian metrics of regularity below $C^2$. These are, on the one hand, the synthetic definition via weak displacement convexity of entropy functionals in the framework of optimal transport, and the distributional one based on non-negativity of the Ricci-tensor in the sense of Schwartz. It turns out that distributional bounds imply entropy bounds for metrics of class $C^1$ and that the converse holds for $C^{1,1}$-metrics under an additional convergence condition on regularisations of the metric.
