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

Xiang-Dong Li

TL;DR

This work extends Perelman’s $W$-entropy framework and Shannon entropy power to the setting of Sturm’s $(K,n,N)$-super Ricci flows on metric measure spaces, including both smooth and synthetic (RCD/mm-space) contexts. It develops time-dependent Witten Laplacians $L_t$, defines $W_{N,K}$ and $W_N$ entropies, and proves their dissipation formulas and monotonicity under the conjugate heat flow, along with a Fatou-like rigidity criterion. The paper then establishes Li-Yau-Hamilton-Perelman Harnack inequalities in this generalized setting, and uses these tools to connect volume non-local collapsing with lower bounds on the $W$-entropy in RCD$(0,N)$ spaces, as well as to derive sharp logarithmic Sobolev inequalities. Overall, the results provide a unified entropy-dissipation framework allowing entropy-based analysis of geometric flows on possibly singular metric measure spaces, broadening the applicability of Perelman’s and Li–Yau–Hamilton techniques to the synthetic setting.

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.

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

TL;DR

This work extends Perelman’s -entropy framework and Shannon entropy power to the setting of Sturm’s -super Ricci flows on metric measure spaces, including both smooth and synthetic (RCD/mm-space) contexts. It develops time-dependent Witten Laplacians , defines and entropies, and proves their dissipation formulas and monotonicity under the conjugate heat flow, along with a Fatou-like rigidity criterion. The paper then establishes Li-Yau-Hamilton-Perelman Harnack inequalities in this generalized setting, and uses these tools to connect volume non-local collapsing with lower bounds on the -entropy in RCD spaces, as well as to derive sharp logarithmic Sobolev inequalities. Overall, the results provide a unified entropy-dissipation framework allowing entropy-based analysis of geometric flows on possibly singular metric measure spaces, broadening the applicability of Perelman’s and Li–Yau–Hamilton techniques to the synthetic setting.

Abstract

In this paper, we extend Perelman's -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 -entropy on RCD spaces. Finally, we use the -entropy to study the logarithmic Sobolev inequality with optimal constant on super Ricci flows on metric measure spaces.
Paper Structure (25 sections, 36 theorems, 303 equations)

This paper contains 25 sections, 36 theorems, 303 equations.

Table of Contents

  1. Introduction
  2. $W$-entropy formulas on smooth super Ricci flows
  3. Smooth $(K, m)$-super Ricci flows
  4. $W$-entropy for $(0, m)$-super Ricci flow
  5. $W$-entropy for $(K, m)$-super Ricci flow
  6. Super Ricci flows on metric measure spaces
  7. Basic facts about RCD spaces
  8. Sturm's super Ricci flows on metric measure spaces
  9. The notion of $(K, n, N)$-super Ricci flows
  10. $W$-entropy formulas on super Ricci flows on mm spaces
  11. $H$-entropy formulas on time dependent metric measure spaces
  12. $W$-entropy formulas on super Ricci flows on mm spaces
  13. $W$-entropy formulas on static RCD spaces
  14. Proofs of theorems
  15. Proof of Theorem \ref{['Th-B1']} and Theorem \ref{['Th-B2']}
  16. ...and 10 more sections

Key Result

Theorem 2.1

(LiLi2015PJM) Let $(M, g(t), \phi(t), t\in [0, T])$ be a compact manifold with family of time dependent metrics and $C^2$-potentials. Suppose that $g(t)$ and $\phi(t)$ satisfy the conjugate equation Let $u={e^{-f}\over (4\pi t)^{m/2}}$ be a positive and smooth solution of the heat equation with initial data $u(0)$ satisfying $\int_M u(0)d\mu(0)=1$. Let Define Then and In particular, if $\{g(

Theorems & Definitions (49)

  • Theorem 2.1
  • Theorem 2.2
  • Definition 3.1
  • Definition 3.2: BS2010
  • Definition 3.3: minimal relaxed gradientAGS2014Invent
  • Definition 3.4: EKS2015KL
  • Theorem 3.5: Theorem 0.7 in Sturm18
  • Definition 3.6
  • Definition 3.7
  • Definition 3.8
  • ...and 39 more