Paper · Nov 1, 2019

Matrix Li-Yau-Hamilton Estimates for Nonlinear Heat Equations

Xin-An Ren

math.DGarXiv:1911.00505
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Abstract

In this paper we are concerned with the matrix Li-Yau-Hamilton estimates for nonlinear heat equations. Firstly, we derive such estimate on a Kähler manifold with a fixed Kähler metric. Then we consider the estimate on Kähler manifolds with Kähler metrics evolving under the rescaled Kähler-Ricci flow. Both of the estimates are generalized to constrained cases. Finally, we extend the estimtes to more general nonlinear heat equations on both Riemannian manifolds and Kähler manifolds.

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Submitted
Nov 1, 2019
Subjects
math.DG, math.CV

Comments: arXiv admin note: substantial text overlap with arXiv:1407.0099

Ricci flowKähler–Ricci flowHarnack inequalitiesKähler geometry

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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 Aronson-Bénilan estimates for the fast diffusion equation under the Ricci flow (Huai-Dong Cao and Meng Zhu, 2016)8Cao 2016 Some new gradient estimates for two nonlinear parabolic equations under Ricci flow (Wen Wang and Hui Zhou, 2017)6Wang 2017 Matrix Inequality for the Laplace Equation (Jiewon Park, 2017)4Park 2017 New differential Harnack inequalities for nonlinear heat equations (Jia-Yong Wu, 2018)3Wu 2018 Harnack inequalities for positive solutions of the heat equation on closed Finsler manifolds (Cyrille Combete, Serge Degla and Leonard Todjihounde, 2018)5Combete 2018 Matrix Li-Yau-Hamilton estimates under Ricci Flow and parabolic frequency (Xiaolong Li and Qi S. Zhang, 2023)2Li 2023 Matrix Li-Yau-Hamilton Estimates under Kähler-Ricci Flow (Xiaolong Li, Hao-Yue Liu and Xin-An Ren, 2023)1Li 2023 Log gradient estimates for heat type equations on manifolds (Qi S. Zhang, 2024)7Zhang 2024
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 Kähler-Ricci Flow

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

    Topics Harnack inequalities, Kähler–Ricci flow, Kähler geometry · Same author Xin-An Ren · Shared terms “Hamilton estimates”, “Yau Hamilton”, “heat equation”

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

    Xiaolong Li and Qi S. Zhang

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

  3. New differential Harnack inequalities for nonlinear heat equations

    Jia-Yong Wu

    Topics Harnack inequalities · Shared terms “nonlinear heat”, “heat equation”, “constrained”

  4. Matrix Inequality for the Laplace Equation

    Jiewon Park

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “heat equation”, “Yau”

  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”, “Yau”

  6. Some new gradient estimates for two nonlinear parabolic equations under Ricci flow

    Wen Wang and Hui Zhou

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

  7. Log gradient estimates for heat type equations on manifolds

    Qi S. Zhang

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

  8. Aronson-Bénilan estimates for the fast diffusion equation under the Ricci flow

    Huai-Dong Cao and Meng Zhu

    Topics Harnack inequalities · Shared terms “Yau Hamilton”, “Hamilton estimates”, “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) 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) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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.