Monotonicity and rigidity of the W-entropy on RCD(0, N) spaces

2018/11/17 by Kazumasa Kuwada, Xiang-Dong Li, Kuwada, Kazumasa +2 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Curvature #Differential Geometry (math.DG) #Entropy (arrow of time) #Euclidean geometry #Euclidean space #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Mathematical analysis #Mathematics #Metric Geometry (math.MG) #Metric space #Monotonic function #Physics #Probability (math.PR) #Pure mathematics #Ricci curvature #Rigidity (electromagnetism) #Upper and lower bounds #math.AP #math.DG #math.MG #math.PR

paper · pdf · doi:10.48550/arxiv.1811.07228

published in arXiv (Cornell University) (Cornell University) · 30 pages

arxiv created 2018/11/17 · openalex publication_date 2018/11/17 · arxiv updated 2018/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

By means of a space-time Wasserstein control, we show the monotonicity of the W-entropy functional in time along heat flows on possibly singular metric measure spaces with non-negative Ricci curvature and a finite upper bound of dimension in an appropriate sense. The associated rigidity result on the rate of dissipation of the W-entropy is also proved. These extend known results even on weighted Riemannian manifolds in some respects. In addition, we reveal that some singular spaces will exhibit the rigidity models while only the Euclidean space does in the class of smooth weighted Riemannian manifolds.

Citations

• Ricci curvature for metric-measure spaces via optimal transport
• On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces
• Metric measure spaces with Riemannian Ricci curvature bounded from below
• The splitting theorem in non-smooth context
• W-entropy formulas and Langevin deformation of flows on Wasserstein space over Riemannian manifolds
• The Globalization Theorem for the Curvature Dimension Condition

Cited by

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

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