Exponential contraction in Wasserstein distance on static and evolving manifolds

Equivalence between dimensional contractions in Wasserstein distance and the curvature-dimension condition

The curvature-dimension condition is a generalization of the Bochner inequality to weighted Riemannian manifolds and general metric measure spaces. It is now known to be equivalent to evolution variational inequalities for the heat semigroup, and quadratic Wasserstein distance contraction properties at different times. On the other hand, in a compact Riemannian manifold, it implies a same-time Wasserstein contraction property for this semigroup. In this work we generalize the latter result to metric measure spaces and more importantly prove the converse: contraction inequalities are equivalent…

关于热流沿程 Wasserstein 收缩的刚性

On the rigidity of Wasserstein contraction along heat flows

本文证明热流 Wasserstein 收缩刚性等价于 Bakry–Émery 梯度估计刚性，并在 RCD 空间与加权流形中推出等号情形的直线分裂及加权欧氏空间刻画。

截面曲率与矩阵位移凸性

Sectional curvature and matrix displacement convexity

该研究证明黎曼流形截面曲率非负等价于熵张量的矩阵位移凸性，还将此结论应用于强化HWI不等式等结果，且内在维数Wasserstein收缩不等式可刻画非负截面曲率

Non-contraction of heat flow on Minkowski spaces

We study contractivity properties of gradient flows for functions on normed spaces or, more generally, on Finsler manifolds. Contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions. This is different from the usual convexity along geodesics in non-Riemannian Finsler manifolds. As an application, we show that the heat flow on Minkowski normed spaces other than inner product spaces is not contractive with respect to the quadratic Wasserstein distance.

Evolution variational inequality and Wasserstein control in variable curvature context

In this note we continue the analysis of metric measure space with variable ricci curvature bounds. First, we study $(κ,N)$-convex functions on metric spaces where $κ$ is a lower semi-continuous function, and gradient flow curves in the sense of a new evolution variational inequality that captures the information that is provided by $κ$. Then, in the spirit of previous work by Erbar, Kuwada and Sturm \cite{erbarkuwadasturm} we introduce an entropic curvature-dimension condition $CD^e(κ,N)$ for metric measure spaces and lower semi-continuous $κ$. This condition is stable with respect to Gromov …

Evolution of Contractions between Non-Compact Manifolds

Let $N$ be a complete manifold with bounded geometry, such that $\sec_N\le -σ< 0$ for some positive constant $σ$. We investigate the mean curvature flow of the graphs of smooth length-decreasing maps $f:\mathbb{R}^m\to N$. In this case, the solution exists for all times and the evolving submanifold stays the graph of a length-decreasing map $f_t$. We further prove uniform decay estimates for all derivatives of order $\ge 2$ of $f_t$ along the flow.