The Wasserstein distance for Ricci shrinkers

Abstract

Let $(M^n,g,f)$ be a Ricci shrinker such that $\textrm{Ric}_f=\frac{1}{2}g$ and the measure induced by the weighted volume element $(4\pi)^{-\frac{n}{2}}e^{-f}dv_{g}$ is a probability measure. Given a point $p\in M$, we consider two probability measures defined in the tangent space $T_pM$, namely the Gaussian measure $\gamma$ and the measure $\overline{\nu}$ induced by the exponential map of $M$ to $p$. In this paper, we prove a result that provides an upper estimate for the Wasserstein distance with respect to the Euclidean metric $g_0$ between the measures $\overline{\nu}$ and $\gamma$, and which also elucidates the rigidity implications resulting from this estimate.