On the rigidity of Wasserstein contraction along heat flows

Abstract

We establish an equivalence between the rigidity of Wasserstein contraction along heat flows and the rigidity of Bakry--Émery gradient estimates for Lipschitz functions. Applying results of Ambrosio--Brué--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.

30 pages. This version extends the results to all curvature bounds and also to RCD spaces