Optimal maps and exponentiation on finite dimensional spaces with Ricci curvature bounded from below

TL;DR

This paper establishes the existence and uniqueness of optimal transport maps on $RCD^*(K,N)$ spaces with absolutely continuous starting measures, and explores how this relates to defining exponential maps in these spaces.

Contribution

It provides the first proof of existence and uniqueness of optimal maps in $RCD^*(K,N)$ spaces, linking optimal transport to the concept of exponential maps in this setting.

Findings

Existence and uniqueness of optimal maps on $RCD^*(K,N)$ spaces.

Connection between optimal transport and exponential maps in these spaces.

Framework for further geometric analysis in metric measure spaces.

Abstract

We prove existence and uniqueness of optimal maps on $RCD^*(K,N)$ spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation.