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

2025/05/06 by Xiangdong Li, Li, Xiang-Dong
Mathematics · Physics and Astronomy · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Probability (math.PR) #Statistical Mechanics and Entropy

paper · pdf · doi:10.48550/arxiv.2505.03202

openalex publication_date 2025/05/06 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28

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.

Citations

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Related

• Entropy Functionals, Sobolev Inequalities and kappa-Noncollapsing Estimates along the Ricci Flow
• W-entropy, super Perelman Ricci flows and (K, m)-Ricci solitons
• On the Shannon entropy power on Riemannian manifolds and Ricci flow
• Entropy and heat kernel bounds on a Ricci flow background
• Harnack inequalities and W-entropy formula for Witten Laplacian on Riemannian manifolds with K-super Perelman Ricci flow