Optimal transportation and monotonic quantities on evolving manifolds

TL;DR

This paper extends optimal transportation theory to evolving manifolds under Ricci flow-like conditions, generalizing monotonicity results for entropy and volume, and unifying several key geometric monotonicity formulas.

Contribution

It adapts Topping's optimal transportation framework to more general evolving manifolds, extending known monotonicity results for entropy and volume under Ricci flow.

Findings

Extended Topping's $ abla$-optimal transportation to general flows

Generalized monotonicity of List's and Perelman's $ abla$-entropy

Reestablished monotonicity of M"uller’s reduced volume

Abstract

In this note we will adapt Topping's $\mathcal{L}$-optimal transportation theory for Ricci flow to a more general situation, i.e. to a closed manifold $(M,g_{ij}(t))$ evolving by $\partial_tg_{ij}=-2S_{ij}$, where $S_{ij}$ is a symmetric tensor field of (2,0)-type on $M$. We extend some recent results of Topping, Lott and Brendle, generalize the monotonicity of List's (and hence also of Perelman's) $\mathcal{W}$-entropy, and recover the monotonicity of M$\ddot{u}$ller's (and hence also of Perelman's) reduced volume.