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 $(4\pi )^{-\frac{n}{2}}e^{-f}dv_{g}$ 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.