Paper · Jul 30, 2024

Log gradient estimates for heat type equations on manifolds

Qi S. Zhang

math.DGarXiv:2407.20719Surveys in Geometric Analysis 2023, edited by Gang Tian, Qing Han, Zhenlei Zhang, Science Press, Beijing 2024, pp186-212
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Abstract

In this short survey paper, we first recall the log gradient estimates for the heat equation on manifolds by Li-Yau, R. Hamilton and later by Perelman in conjunction with the Ricci flow. Then we will discuss some of their applications and extensions focusing on sharp constants and improved curvature conditions.

Comments: a survey paper published in a book chapter, 23 pages

Ricci flowHarnack inequalities

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 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 Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with evolving metrics and potentials (Ali Taheri and Vahideh Vahidifar, 2023)4Taheri 2023 Matrix Li-Yau-Hamilton estimates under Ricci Flow and parabolic frequency (Xiaolong Li and Qi S. Zhang, 2023)2Li 2023 A direct approach to sharp Li-Yau Estimates on closed manifolds with negative Ricci lower bound (Xingyu Song, Ling Wu and Meng Zhu, 2023)5Song 2023 Parabolic frequency monotonicity for two nonlinear equations under Ricci flow (Chuanhuan Li, Yi Li, Kairui Xu et al., 2023)7Li 2023 Li-Yau Estimates for a Nonlinear Parabolic Equation on Finsler Manifolds (Bin Shen and Yuhan Zhu, 2023)6Shen 2023 Gradient estimation of a generalized non-linear heat type equation along Super-Perelman Ricci flow on weighted Riemannian manifolds (Yanlin Li, Abimbola Abolarinwa, Suraj Ghosh et al., 2024)1Li 2024 Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow (Juanling Lu and Yu Zheng, 2025)3Lu 2025 Monotonicity of Perelman 𝓦-Entropy of Mean Curvature Flow (Xiang-Dong Li and Qi Yan, 2025)8Li 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Gradient estimation of a generalized non-linear heat type equation along Super-Perelman Ricci flow on weighted Riemannian manifolds

    Yanlin Li, Abimbola Abolarinwa, Suraj Ghosh et al. PDF

    Topics Harnack inequalities · Shared terms “heat type”, “type equations”, “Perelman”

  2. Matrix Li-Yau-Hamilton estimates under Ricci Flow and parabolic frequency

    Xiaolong Li and Qi S. Zhang PDF

    Topics Harnack inequalities · Same author Qi S. Zhang · Shared terms “heat equation”, “Hamilton”, “Yau”

  3. Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow

    Juanling Lu and Yu Zheng PDF

    Topics Harnack inequalities · Shared terms “gradient estimate”, “heat equation”, “curvature condition”

  4. Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with evolving metrics and potentials

    Ali Taheri and Vahideh Vahidifar PDF

    Topics Harnack inequalities · Shared terms “gradient estimate”, “Perelman”

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

    Xingyu Song, Ling Wu and Meng Zhu PDF

    Topics Harnack inequalities · Shared terms “gradient estimate”, “heat equation”, “Yau”

  6. Li-Yau Estimates for a Nonlinear Parabolic Equation on Finsler Manifolds

    Bin Shen and Yuhan Zhu PDF

    Topics Harnack inequalities · Shared terms “gradient estimate”, “Yau”

  7. Parabolic frequency monotonicity for two nonlinear equations under Ricci flow

    Chuanhuan Li, Yi Li, Kairui Xu et al. PDF

    Topics Harnack inequalities

  8. Monotonicity of Perelman -Entropy of Mean Curvature Flow

    Xiang-Dong Li and Qi Yan PDF

    Topics Harnack inequalities · Shared terms “Perelman”, “Hamilton”

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) 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)

In the guidebook

  1. 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.

  2. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  3. 12A.6
    𝓛-Geodesics and the Variational FormulasBook 12A · Entropy and Noncollapsing

    Perelman's §7.1 derived line by line: the first and second variation of 𝓛-length, 𝓛-Jacobi fields, adapted fields, Hamilton's Harnack expressions, and the differential inequalities for the reduced distance.