Paper · Jan 6, 2017

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

Wen Wang, Hui Zhou

math.DGarXiv:1701.01651
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Abstract

In this paper, by maximum principle and cutoff function, we investigate gradient estimates for positive solutions to two nonlinear parabolic equations under Ricci flow. The related Harnack inequalities are deduced. An result about positive solutions on closed manifolds under Ricci flow is abtained. As applications, gradient estimates and Harnack inequalities for positive solutions to the heat equation under Ricci flow are derived. These results in the paper can be regard as generalizing the gradient estimates of Li-Yau, J. Y. Li, Hamilton and Li-Xu to the Ricci flow. Our results also improve the estimates of S. P. Liu and J. Sun to the nonlinear parabolic equation under Ricci flow.

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Submitted
Jan 6, 2017
Subjects
math.DG
Ricci flowHarnack inequalities

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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 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 Local Aronson-Benolan type gradient estimates for the porous medium type equation under Ricci (Wen Wang, Hui Zhou and Dapeng Xia, 2017)1Wang 2017 New differential Harnack inequalities for nonlinear heat equations (Jia-Yong Wu, 2018)8Wu 2018 Harnack inequalities for positive solutions of the heat equation on closed Finsler manifolds (Cyrille Combete, Serge Degla and Leonard Todjihounde, 2018)3Combete 2018 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 Parabolic frequency monotonicity for two nonlinear equations under Ricci flow (Chuanhuan Li, Yi Li, Kairui Xu et al., 2023)2Li 2023 Li-Yau Estimates for a Nonlinear Parabolic Equation on Finsler Manifolds (Bin Shen and Yuhan Zhu, 2023)5Shen 2023 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)6Lu 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Local Aronson-Benolan type gradient estimates for the porous medium type equation under Ricci

    Wen Wang, Hui Zhou and Dapeng Xia

    Topics Harnack inequalities · Same authors Wen Wang, Hui Zhou · Shared terms “gradient estimates”, “related Harnack”, “positive solutions”

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

    Chuanhuan Li, Yi Li, Kairui Xu et al.

    Topics Harnack inequalities · Shared terms “nonlinear parabolic”, “parabolic equations”, “positive solutions”

  3. 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 “positive solutions”, “gradient estimates”, “heat equation”

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

    Ali Taheri and Vahideh Vahidifar

    Topics Harnack inequalities · Shared terms “nonlinear parabolic”, “parabolic equations”, “gradient estimates”

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

    Bin Shen and Yuhan Zhu

    Topics Harnack inequalities · Shared terms “nonlinear parabolic”, “parabolic equations”, “positive solutions”

  6. 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”, “gradient estimates”, “heat equation”

  7. Log gradient estimates for heat type equations on manifolds

    Qi S. Zhang

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

  8. New differential Harnack inequalities for nonlinear heat equations

    Jia-Yong Wu

    Topics Harnack inequalities · Shared terms “heat equation”, “closed manifold”

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.