Paper · Oct 10, 2020

Dimension-free Harnack inequalities for conjugate heat equations and their applications to geometric flows

Li-Juan Cheng, Anton Thalmaier

math.PRarXiv:2010.04916
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Abstract

Let be a differentiable manifold endowed with a family of complete Riemannian metrics evolving under a geometric flow over the time interval . In this article, we give a probabilistic representation for the derivative of the corresponding conjugate semigroup on which is generated by a Schrödinger type operator. With the help of this derivative formula, we derive fundamental Harnack type inequalities in the setting of evolving Riemannian manifolds. In particular, we establish a dimension-free Harnack inequality and show how it can be used to achieve heat kernel upper bounds in the setting of moving metrics. Moreover, by means of the supercontractivity of the conjugate semigroup, we obtain a family of canonical log-Sobolev inequalities. We discuss and apply these results both in the case of the so-called modified Ricci flow and in the case of general geometric flows.

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Submitted
Oct 10, 2020
Subjects
math.PR, math.DG
Journal
Analysis & PDE 16 (2023) 1589-1620
DOI
10.2140/apde.2023.16.1589
Ricci flowPerelman entropyHarnack inequalitiesHeat kernel

Its neighbourhood

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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 2016, 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 On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows (Songzi Li and Xiang-Dong Li, 2017)3Li 2017 W-entropy, super Perelman Ricci flows and (K, m)-Ricci solitons (Songzi Li and Xiang-Dong Li, 2017)8Li 2017 Harnack inequalities on totally geodesic foliations with transverse Ricci flow (Qi Feng, 2017)1Feng 2017 Harnack inequalities for positive solutions of the heat equation on closed Finsler manifolds (Cyrille Combete, Serge Degla and Leonard Todjihounde, 2018)5Combete 2018 Hessian estimates for the conjugate heat equation coupled with the Ricci flow (Hong Huang, 2021)6Huang 2021 A direct approach to sharp Li-Yau Estimates on closed manifolds with negative Ricci lower bound (Xingyu Song, Ling Wu and Meng Zhu, 2023)4Song 2023 Sharp Gaussian Isoperimetry along a Ricci Flow (Robert Koirala, 2026)7Koirala 2026 Pseudolocality of G₂-Laplacian Flow (Weiyong He and Longhuan Yan, 2026)2He 2026
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  1. Harnack inequalities on totally geodesic foliations with transverse Ricci flow

    Qi Feng

    Topics Heat kernel, Harnack inequalities · Shared terms “kernel upper”, “upper bound”

  2. Pseudolocality of -Laplacian Flow

    Weiyong He and Longhuan Yan

    Topics Heat kernel, Perelman entropy, Harnack inequalities · Shared terms “geometric flows”, “conjugate heat”, “Harnack type”

  3. On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows

    Songzi Li and Xiang-Dong Li

    Topics Harnack inequalities · Shared terms “free Harnack”, “dimension free”, “heat equation”

  4. A direct approach to sharp Li-Yau Estimates on closed manifolds with negative Ricci lower bound

    Xingyu Song, Ling Wu and Meng Zhu

    Topics Heat kernel, Harnack inequalities · Shared terms “conjugate heat”, “Riemannian manifold”

  5. Harnack inequalities for positive solutions of the heat equation on closed Finsler manifolds

    Cyrille Combete, Serge Degla and Leonard Todjihounde

    Topics Harnack inequalities · Shared terms “heat equation”, “Harnack type”, “type inequality”

  6. Hessian estimates for the conjugate heat equation coupled with the Ricci flow

    Hong Huang

    Topics Heat kernel · Shared terms “conjugate heat”, “upper bound”

  7. Sharp Gaussian Isoperimetry along a Ricci Flow

    Robert Koirala

    Topics Heat kernel, Perelman entropy · Shared terms “conjugate heat”, “log Sobolev”, “Sobolev inequality”

  8. -entropy, super Perelman Ricci flows and -Ricci solitons

    Songzi Li and Xiang-Dong Li

    Topics Harnack inequalities · Shared terms “free Harnack”, “dimension free”, “heat equation”

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) Perelman entropy Perelman found that Ricci flow is a gradient flow once a measure is fixed. His 𝓕-functional and 𝓦-entropy increase along the flow; a lower bound… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Harnack inequalities Differential Harnack inequalities compare a solution with itself at different points and times. Hamilton's matrix Harnack inequality says curvature… Start with: On the parabolic kernel of the Schrödinger operator (Li et al., 1986) 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. 6D.5
    Diffusions on Manifolds and CurvatureBook 6D · Diffusions, Finance and Computation

    Brownian motion on a Riemannian manifold, the Bakry–Émery calculus, curvature-dimension inequalities, gradient estimates and log-Sobolev, coupling, and Brownian motion along a Ricci flow.