On the rigidity of Wasserstein contraction along heat flows

TL;DR

This paper links the rigidity of Wasserstein contraction along heat flows to gradient estimates, showing that sharp contraction implies the space splits off a line, with applications to $ cd$ spaces and weighted manifolds.

Contribution

It establishes a new equivalence between Wasserstein contraction rigidity and gradient estimate rigidity, and characterizes spaces with sharp contraction for all point pairs.

Findings

Rigidity of Wasserstein contraction implies space splits off a line.

Characterization of weighted Euclidean spaces with sharp Wasserstein contraction.

Extension of rigidity results to all curvature bounds $K \

Abstract

We establish an equivalence between the rigidity of Wasserstein contraction along heat flows and the rigidity of Bakry--\'Emery gradient estimates for Lipschitz functions. Applying results of Ambrosio--Bru\'e--Semola and Han, we show that if an $\rcd$ space with Ricci lower bound $K\in[0,\infty)$ admits two distinct points $x,y$ such that the $2$-Wasserstein distance between the associated heat kernels satisfies \[ W_2(p_t(x,\cdot), p_t(y,\cdot)) = e^{-Kt} d(x,y), \] then the space splits off a line. Moreover, for weighted smooth manifolds, we provide a direct proof of the rigidity theorem for all curvature bounds $K \in \mathbb{R}$. In particular, we characterize a class of weighted Euclidean spaces as the only spaces where the Wasserstein contraction is sharp for all pairs of points.