On Perelman's $W$-entropy and Shannon entropy power for super Ricci flows on metric measure spaces

TL;DR

This paper generalizes Perelman's $W$-entropy and Shannon entropy power to super Ricci flows on metric measure spaces, establishing key inequalities and applications in non-smooth geometric analysis.

Contribution

It extends entropy formulas and inequalities from smooth to non-smooth spaces, linking volume collapse and entropy bounds in metric measure spaces.

Findings

Proves the concavity of Shannon entropy power for super Ricci flows.

Establishes the Li-Yau-Hamilton-Perelman Harnack inequality in this setting.

Shows the equivalence between volume non-collapsing and $W$-entropy boundedness.

Abstract

In this paper, we extend Perelman's $W$-entropy formula and the concavity of the Shannon entropy power from smooth Ricci flow to super Ricci flows on metric measure spaces. Moreover, we prove the Li-Yau-Hamilton-Perelman Harnack inequality on super Ricci flows. As a significant application, we prove the equivalence between the volume non-local collapsing property and the lower boundedness of the $W$-entropy on RCD$(0, N)$ spaces. Finally, we use the $W$-entropy to study the logarithmic Sobolev inequality with optimal constant on super Ricci flows on metric measure spaces.