Paper · Jul 8, 2023

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

Xingyu Song, Ling Wu, Meng Zhu

math.DGarXiv:2307.03879v2
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Abstract

Recently, Qi S.Zhang [26] has derived a sharp Li-Yau estimate for positive solutions of the heat equation on closed Riemannian manifolds with the Ricci curvature bounded below by a negative constant. The proof is based on an integral iteration argument which utilizes Hamilton's gradient estimate, heat kernel Gaussian bounds and parabolic Harnack inequality. In this paper, we show that the sharp Li-Yau estimate can actually be obtained directly following the classical maximum principle argument, which simplifies the proof in [26]. In addition, we apply the same idea to the heat and conjugate heat equations under the Ricci flow and prove some Li-Yau type estimates with optimal coefficients.

Comments: 14 pages

Ricci flowHarnack inequalitiesHeat kernel

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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.

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 Hessian estimates for the conjugate heat equation coupled with the Ricci flow (Hong Huang, 2021)4Huang 2021 Matrix Li-Yau-Hamilton estimates under Ricci Flow and parabolic frequency (Xiaolong Li and Qi S. Zhang, 2023)1Li 2023 Matrix Li-Yau-Hamilton Estimates under Kähler-Ricci Flow (Xiaolong Li, Hao-Yue Liu and Xin-An Ren, 2023)2Li 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 Log gradient estimates for heat type equations on manifolds (Qi S. Zhang, 2024)5Zhang 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 Pseudolocality of G₂-Laplacian Flow (Weiyong He and Longhuan Yan, 2026)8He 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Matrix Li-Yau-Hamilton estimates under Ricci Flow and parabolic frequency

    Xiaolong Li and Qi S. Zhang PDF

    Topics Heat kernel, Harnack inequalities · Shared terms “Yau”, “positive solutions”, “conjugate heat”

  2. Matrix Li-Yau-Hamilton Estimates under Kähler-Ricci Flow

    Xiaolong Li, Hao-Yue Liu and Xin-An Ren PDF

    Topics Heat kernel, Harnack inequalities · Shared terms “Yau”, “positive solutions”, “conjugate heat”

  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 “heat equation”, “positive solutions”, “gradient estimate”

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

    Hong Huang PDF

    Topics Heat kernel · Shared terms “conjugate heat”, “positive solutions”

  5. Log gradient estimates for heat type equations on manifolds

    Qi S. Zhang 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 “Yau estimate”, “positive solutions”, “gradient estimate”

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

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

    Topics Harnack inequalities · Shared terms “closed manifolds”, “positive solutions”, “lower bound”

  8. Pseudolocality of -Laplacian Flow

    Weiyong He and Longhuan Yan PDF

    Topics Heat kernel, Harnack inequalities · Shared terms “conjugate heat”, “lower bound”

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) 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. 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. 12A.9
    PseudolocalityBook 12A · Entropy and Noncollapsing

    Almost-Euclidean regions stay controlled, and the Harnack inequality behind it.

  3. 12A.10
    The Proof of PseudolocalityBook 12A · Entropy and Noncollapsing

    Perelman's §10 in detail: the point-picking argument, the conjugate heat kernel at the bad point, the localised entropy, and the contradiction with the almost-Euclidean isoperimetric inequality.