The Wasserstein distance for Ricci shrinkers

TL;DR

This paper investigates the Wasserstein distance between two probability measures associated with Ricci shrinkers, providing upper bounds and exploring the rigidity implications of these estimates.

Contribution

It introduces an upper estimate for the Wasserstein distance between measures on Ricci shrinkers and discusses the rigidity consequences of this estimate.

Findings

Established an upper bound for the Wasserstein distance between measures on Ricci shrinkers.

Linked the distance estimate to rigidity properties of Ricci shrinkers.

Enhanced understanding of measure behavior in the geometric analysis of 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.