Paper · Dec 11, 2022

Nash Entropy and Heat Kernel on Generalized Ricci Flow

Xilun Li

math.DGarXiv:2212.05414v2
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Abstract

We introduce analogous geometric quantities and prove some geometric and analytic bounds in [Bam20a] to generalized Ricci flow.

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Submitted
Dec 11, 2022
Last revised
Nov 6, 2023 (v2)
Subjects
math.DG

Comments: 17 pages, comments are welcome

Ricci flowGeneralized & G₂ flowsPerelman entropyHeat kernel

Its neighbourhood

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.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Entropy and heat kernel bounds on a Ricci flow background (Richard H Bamler, 2020)1Bamler 2020 Scalar curvature, entropy, and generalized Ricci flow (Jeffrey Streets, 2022)4Streets 2022 Heat kernel on Ricci shrinkers (II) (Yu Li and Bing Wang, 2023)6Li 2023 On the shrinking solitons of generalized Ricci flow (Xilun Li and Yanan Ye, 2024)8Li 2024 Perelman's entropy and heat kernel bounds on RCD spaces (Camillo Brena, 2025)7Brena 2025 The Fisher Metric of the Ricci Flow Heat Kernel (Bennett Chow and Robert Koirala, 2026)3Chow 2026 Heat kernel on Ricci shrinker metric measure spaces (Bing Wang and Jie Wang, 2026)5Wang 2026 Pseudolocality of G₂-Laplacian Flow (Weiyong He and Longhuan Yan, 2026)2He 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Entropy and heat kernel bounds on a Ricci flow background

    Richard H BamlerThe canon

    Topics Heat kernel, Perelman entropy · Shared terms “Nash entropy”, “analytic bounds”

  2. Pseudolocality of -Laplacian Flow

    Weiyong He and Longhuan Yan

    Topics Heat kernel, Perelman entropy, Generalized & G₂ flows · Shared terms “Nash entropy”

  3. The Fisher Metric of the Ricci Flow Heat Kernel

    Bennett Chow and Robert Koirala

    Topics Heat kernel, Perelman entropy · Shared terms “Nash entropy”

  4. Scalar curvature, entropy, and generalized Ricci flow

    Jeffrey Streets

    Topics Heat kernel, Generalized & G₂ flows · Shared terms “Nash entropy”

  5. Heat kernel on Ricci shrinker metric measure spaces

    Bing Wang and Jie Wang

    Topics Heat kernel, Perelman entropy · Shared terms “Nash entropy”

  6. Heat kernel on Ricci shrinkers (II)

    Yu Li and Bing Wang

    Topics Heat kernel

  7. Perelman's entropy and heat kernel bounds on RCD spaces

    Camillo Brena

    Topics Heat kernel, Perelman entropy

  8. On the shrinking solitons of generalized Ricci flow

    Xilun Li and Yanan Ye

    Topics Generalized & G₂ flows · Same author Xilun Li

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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) Perelman entropy Perelman found that Ricci flow is a gradient flow. His 𝓕-functional and 𝓦-entropy increase along the flow, and their monotonicity rules out… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Heat kernel The heat kernel of the conjugate heat equation, coupled to Ricci flow, is the probe Perelman and Bamler use to measure geometry. Start with: On the parabolic kernel of the Schrödinger operator (Li et al., 1986)

In the guidebook

  1. 6A.12
    Entropy, Information and DiffusionBook 6A · The Heat Equation and Its Relatives

    Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.

  2. 6D.4
    Fokker–Planck, Time Reversal and Diffusion ModelsBook 6D · Diffusions, Finance and Computation

    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.

  3. 6A.3
    The Heat Equation on ℝⁿBook 6A · The Heat Equation and Its Relatives

    The heat kernel, smoothing, uniqueness, Brownian motion, and a Gaussian that runs to Perelman.