Heat Flows on Time-dependent Metric Measure Spaces and Super-Ricci Flows

2016/11/08 by Eva Kopfer, Kopfer, Eva, Karl‐Theodor Sturm +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1611.02570

openalex publication_date 2016/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the heat equation on time-dependent metric measure spaces (as well as the dual and the adjoint heat equation) and prove existence, uniqueness and regularity. Of particular interest are properties which characterize the underlying space as a super Ricci flow as previously introduced by the second author. Our main result yields the equivalence of (i) dynamic convexity of the Boltzmann entropy on the (time-dependent) L²-Wasserstein space; (ii) monotonicity of L²-Kantorovich-Wasserstein distances under the dual heat flow acting on probability measures (backward in time); (iii) gradient estimates for the heat flow acting on functions (forward in time); (iv) a Bochner inequality involving the time-derivative of the metric. Moreover, we characterize the heat flow on functions as the unique forward EVI-flow for the (time-dependent) energy in L²-Hilbert space and the dual heat flow on probability measures as the unique backward EVI-flow for the (time-dependent) Boltzmann entropy in L²-Wasserstein space.

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
• Ricci curvature of Markov chains on metric spaces
• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Ricci flow with surgery on three-manifolds

Cited by

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

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• Super-Ricci Flows for Metric Measure Spaces
• Synthetic notions of Ricci flow for metric measure spaces
• Heat flow and calculus on metric measure spaces with Ricci curvature bounded below - the compact case
• Heat flow on time-dependent manifolds
• Metric measure spaces with Riemannian Ricci curvature bounded from below