As on arXiv
Abstract
As a metric measure space possessing positive Bakry-Émery curvature, a Ricci shrinker with potential function admits a heat kernel under the weighted volume measure . In this paper, we study systematically and establish a series of fundamental estimates. Moreover, we clarify the equivalence between and the spacetime heat kernel under Ricci flow induced by a Ricci shrinker. Inspired by this relation, we extend corresponding analysis tools of Ricci flows to Ricci shrinker metric measure spaces, such as Nash entropy and reduced distance. As a direct application, we prove that implies .
Comments: 47 pages
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Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
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Heat kernel on Ricci shrinkers (II)
Topics Heat kernel, Limit spaces & RCD, Ricci solitons · Same author Bing Wang · Shared terms “Ricci shrinkers”, “flow induced”
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Nash Entropy and Heat Kernel on Generalized Ricci Flow
Topics Heat kernel, Perelman entropy · Shared terms “Nash entropy”
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Perelman's entropy and heat kernel bounds on RCD spaces
Topics Heat kernel, Perelman entropy, Limit spaces & RCD · Shared terms “Bakry Émery”
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Entropy and heat kernel bounds on a Ricci flow background
Topics Heat kernel, Perelman entropy, Limit spaces & RCD · Shared terms “Nash entropy”
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The Fisher Metric of the Ricci Flow Heat Kernel
Topics Heat kernel, Perelman entropy · Shared terms “Nash entropy”
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Rigidity of the round cylinders in Ricci shrinkers
Topics Limit spaces & RCD, Ricci solitons · Same author Bing Wang · Shared terms “Ricci shrinkers”
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The Ricci-DeTurck flow on complete manifolds
Topics Heat kernel · Same author Bing Wang
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Non-collapsing of Ricci shrinkers with bounded curvature
Topics Limit spaces & RCD, Ricci solitons · Shared terms “Ricci shrinkers”, “metric measure”, “measure spaces”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 6A.12Entropy, Information and Diffusion
Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.
- 6D.4Fokker–Planck, Time Reversal and Diffusion Models
How densities of diffusions evolve, relative entropy and convergence to equilibrium, reversing a diffusion in time with the score, and the generative models built on it, with Perelman's conjugate heat equation as the geometric twin.
- 6C.11Optimal Transport and Ricci Curvature
Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.