Paper · Dec 14, 2023

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

Bin Shen, Yuhan Zhu

math.DGarXiv:2402.00006v2
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Abstract

In this paper, we explore the positive solutions to the Finslerian nonlinear equation which is related to Ricci solitons and serves as the Euler-Lagrange equation to the Finslerian log-energy functional. We then obtain the global gradient estimate of its positive solution on a compact Finsler metric measure space with the weighted Ricci curvature bounded below. Furthermore, using a new comparison theorem developed by the first author, we also establish a local gradient estimate on a non-compact forward complete Finsler metric measure spaces with the mixed weighted Ricci curvature bounded below, as well as finite bounds of misalignment and some non-Riemannian curvatures. Lastly, we prove the Harnack inequalities and a Liouville-type theorem of such solutions.

Ricci solitonsHarnack inequalitiesLimit spaces & RCDBoundaries & non-compact

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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 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 Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with evolving metrics and potentials (Ali Taheri and Vahideh Vahidifar, 2023)2Taheri 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)4Song 2023 Parabolic frequency monotonicity for two nonlinear equations under Ricci flow (Chuanhuan Li, Yi Li, Kairui Xu et al., 2023)3Li 2023 Log gradient estimates for heat type equations on manifolds (Qi S. Zhang, 2024)6Zhang 2024 Bounds of Scalar curvature, S-curvature and distortion on ∞-Einstein Finsler manifolds (Bin Shen, 2024)1Shen 2024 On Perelman's W-entropy and Shannon entropy power for super Ricci flows on metric measure spaces (Xiang-Dong Li, 2025)7Li 2025 Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow (Juanling Lu and Yu Zheng, 2025)5Lu 2025 Fixed geodesics of involutions on 3-spheres (Bin Shen and Hui Zhang, 2026)8Shen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Bounds of Scalar curvature, S-curvature and distortion on -Einstein Finsler manifolds

    Bin Shen PDF

    Topics Limit spaces & RCD, Ricci solitons · Same author Bin Shen · Shared terms “Finsler manifolds”, “Finsler metric”, “metric measure”

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

    Ali Taheri and Vahideh Vahidifar PDF

    Topics Harnack inequalities, Limit spaces & RCD · Shared terms “nonlinear parabolic”, “parabolic equations”, “metric measure”

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

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

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

  4. 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 Harnack inequalities · Shared terms “Yau estimate”, “positive solutions”, “gradient estimate”

  5. 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 “positive solutions”, “gradient estimate”, “curvature bounded”

  6. Log gradient estimates for heat type equations on manifolds

    Qi S. Zhang PDF

    Topics Harnack inequalities · Shared terms “gradient estimate”, “Yau”

  7. On Perelman's -entropy and Shannon entropy power for super Ricci flows on metric measure spaces

    Xiang-Dong Li PDF

    Topics Harnack inequalities, Limit spaces & RCD · Shared terms “metric measure”, “measure spaces”, “Yau”

  8. Fixed geodesics of involutions on -spheres

    Bin Shen and Hui Zhang PDF

    Topics Limit spaces & RCD, Ricci solitons · Same author Bin Shen · Shared terms “comparison”

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 solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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.