Paper · Mar 4, 2025

Perelman's entropy and heat kernel bounds on RCD spaces

Camillo Brena

math.DGarXiv:2503.03017
View PDF View HTML arXiv page

As on arXiv

Abstract

We study Perelman's W-entropy functional on finite-dimensional RCD spaces, a synthetic generalization of spaces with Bakry-Émery Ricci curvature bounded from below. We rigorously justify the formula for the time derivative of the W-entropy and derive its monotonicity and rigidity properties. Additionally, we establish bounds for solutions of the heat equation, which are of independent interest.

View PDF View HTML arXiv page
Submitted
Mar 4, 2025
Subjects
math.DG
Perelman entropyHeat kernelWeak & singular flowsLimit spaces & RCDStability & uniqueness

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 Ricci limit flows and weak solutions (Beomjun Choi and Robert Haslhofer, 2021)5Choi 2021 Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow (Keita Kunikawa and Yohei Sakurai, 2022)8Kunikawa 2022 Nash Entropy and Heat Kernel on Generalized Ricci Flow (Xilun Li, 2022)2Li 2022 Heat kernel on Ricci shrinkers (II) (Yu Li and Bing Wang, 2023)3Li 2023 Sharp Gaussian Isoperimetry along a Ricci Flow (Robert Koirala, 2026)7Koirala 2026 The Fisher Metric of the Ricci Flow Heat Kernel (Bennett Chow and Robert Koirala, 2026)6Chow 2026 Heat kernel on Ricci shrinker metric measure spaces (Bing Wang and Jie Wang, 2026)4Wang 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, Limit spaces & RCD · Shared terms “kernel bounds”, “curvature bounded”

  2. Nash Entropy and Heat Kernel on Generalized Ricci Flow

    Xilun Li

    Topics Heat kernel, Perelman entropy

  3. Heat kernel on Ricci shrinkers (II)

    Yu Li and Bing Wang

    Topics Heat kernel, Limit spaces & RCD

  4. Heat kernel on Ricci shrinker metric measure spaces

    Bing Wang and Jie Wang

    Topics Heat kernel, Perelman entropy, Limit spaces & RCD · Shared terms “Bakry Émery”

  5. Ricci limit flows and weak solutions

    Beomjun Choi and Robert Haslhofer

    Topics Heat kernel, Limit spaces & RCD, Weak & singular flows · Shared terms “kernel bounds”

  6. The Fisher Metric of the Ricci Flow Heat Kernel

    Bennett Chow and Robert Koirala

    Topics Heat kernel, Perelman entropy, Stability & uniqueness

  7. Sharp Gaussian Isoperimetry along a Ricci Flow

    Robert Koirala

    Topics Heat kernel, Perelman entropy, Stability & uniqueness

  8. Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow

    Keita Kunikawa and Yohei Sakurai

    Topics Heat kernel, Weak & singular flows

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.

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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

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. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    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.