The Wasserstein distance for Ricci shrinkers

Abstract

Let $(M^n,g,f)$ be a Ricci shrinker such that ${\text{Ric}}_f=\frac{1}{2}g$ and the measure induced by the weighted volume element $(4\pi {)}^{-\frac{n}{2}}{e}^{-f}d{v}_g$ is a probability measure. Given a point $p\in M$, we consider two probability measures defined in the tangent space $T_{p}M$, 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.