Paper · Nov 6, 2018

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

Cyrille Combete, Serge Degla, Leonard Todjihounde

math.DGarXiv:1811.02100
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Abstract

The main goal of this paper is to generalize some Li-Yau type gradient estimates to Finsler geometry in order to derive Harnack type inequalities. Moreover, we obtain, under some curvature assumption, a general gradient estimate for positive solutions of the heat equation when the manifold evolving along the Finsler Ricci flow.

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Submitted
Nov 6, 2018
Subjects
math.DG

Comments: 13 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 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 Some new gradient estimates for two nonlinear parabolic equations under Ricci flow (Wen Wang and Hui Zhou, 2017)1Wang 2017 Local Aronson-Benolan type gradient estimates for the porous medium type equation under Ricci (Wen Wang, Hui Zhou and Dapeng Xia, 2017)5Wang 2017 Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds (Songzi Li and Xiang-Dong Li, 2017)8Li 2017 New differential Harnack inequalities for nonlinear heat equations (Jia-Yong Wu, 2018)4Wu 2018 Li-Yau Estimates for a Nonlinear Parabolic Equation on Finsler Manifolds (Bin Shen and Yuhan Zhu, 2023)3Shen 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)6Li 2024 Log gradient estimates for heat type equations on manifolds (Qi S. Zhang, 2024)7Zhang 2024 Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow (Juanling Lu and Yu Zheng, 2025)2Lu 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Some new gradient estimates for two nonlinear parabolic equations under Ricci flow

    Wen Wang and Hui Zhou

    Topics Harnack inequalities · Shared terms “positive solutions”, “gradient estimates”, “heat equation”

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

    Juanling Lu and Yu Zheng

    Topics Harnack inequalities · Shared terms “positive solutions”, “heat equation”, “gradient estimates”

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

    Bin Shen and Yuhan Zhu

    Topics Harnack inequalities · Shared terms “Finsler manifolds”, “positive solutions”, “gradient estimates”

  4. New differential Harnack inequalities for nonlinear heat equations

    Jia-Yong Wu

    Topics Harnack inequalities · Shared terms “heat equation”

  5. Local Aronson-Benolan type gradient estimates for the porous medium type equation under Ricci

    Wen Wang, Hui Zhou and Dapeng Xia

    Topics Harnack inequalities · Shared terms “type gradient”, “gradient estimates”, “positive solutions”

  6. 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.

    Topics Harnack inequalities · Shared terms “positive solutions”, “evolving along”, “manifold evolving”

  7. Log gradient estimates for heat type equations on manifolds

    Qi S. Zhang

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

  8. Hamilton differential Harnack inequality and -entropy for Witten Laplacian on Riemannian manifolds

    Songzi Li and Xiang-Dong Li

    Topics Harnack inequalities · Shared terms “positive solutions”, “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) 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.