Paper · Jul 20, 2023

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

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

math.DGarXiv:2307.10920
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Abstract

We prove matrix Li-Yau-Hamilton estimates for positive solutions to the heat equation and the backward conjugate heat equation, both coupled with the Kähler-Ricci flow. As an application, we obtain a monotonicity formula.

Comments: 13 pages. Comments are welcome

Ricci flowKähler–Ricci flowHarnack inequalitiesHeat kernelKähler geometry

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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)3Huang 2021 Matrix Li-Yau-Hamilton estimates under Ricci Flow and parabolic frequency (Xiaolong Li and Qi S. Zhang, 2023)1Li 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)2Song 2023 A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds (Wangjian Jian, Jian Song and Gang Tian, 2023)7Jian 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)4Lu 2025 Harnack Inequality for f-Mean Curvature Flow (Xiang-Dong Li and Qi Yan, 2025)6Li 2025 Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits (Lei Zhang and Zhenlei Zhang, 2026)8Zhang 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 · Same author Xiaolong Li · Shared terms “Hamilton estimates”, “Yau Hamilton”, “backward conjugate”

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

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

    Hong Huang PDF

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

  4. 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”, “Hamilton”

  5. Log gradient estimates for heat type equations on manifolds

    Qi S. Zhang PDF

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

  6. Harnack Inequality for -Mean Curvature Flow

    Xiang-Dong Li and Qi YanWider flows PDF

    Topics Harnack inequalities · Shared terms “Yau Hamilton”

  7. A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds

    Wangjian Jian, Jian Song and Gang Tian PDF

    Topics Harnack inequalities, Kähler–Ricci flow, Kähler geometry

  8. Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits

    Lei Zhang and Zhenlei Zhang PDF

    Topics Heat kernel, Kähler–Ricci flow, Kähler geometry · Shared terms “conjugate heat”

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) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) 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.