Paper · Mar 10, 2023

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

Ali Taheri, Vahideh Vahidifar

math.AParXiv:2303.05802
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Abstract

This article presents new parabolic and elliptic type gradient estimates for positive smooth solutions to a nonlinear parabolic equation involving the Witten Laplacian in the context of smooth metric measure spaces. The metric and potential here are time dependent and evolve under a super Perelman-Ricci flow. The estimates are derived under natural lower bounds on the associated generalised Bakry-Émery Ricci curvature tensors and are utilised in establishing fairly general local and global bounds, Harnack-type inequalities and Liouville-type global constancy theorems to mention a few. Other implications and consequences of the results are also discussed.

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Submitted
Mar 10, 2023
Subjects
math.AP
Journal
Nonlinear Analysis 2023

Comments: 41 pages

Ricci flowHarnack inequalitiesLimit spaces & RCD

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 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 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)1Shen 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)4Li 2024 Metric Measure Spaces and Synthetic Ricci Bounds – Fundamental Concepts and Recent Developments (Karl-Theodor Sturm, 2024)5Sturm 2024 Log gradient estimates for heat type equations on manifolds (Qi S. Zhang, 2024)6Zhang 2024 Synthetic notions of Ricci flow for metric measure spaces (Matthias Erbar, Zhenhao Li and Timo Schultz, 2025)8Erbar 2025 On Perelman's W-entropy and Shannon entropy power for super Ricci flows on metric measure spaces (Xiang-Dong Li, 2025)3Li 2025 Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow (Juanling Lu and Yu Zheng, 2025)7Lu 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Li-Yau Estimates for a Nonlinear Parabolic Equation on Finsler Manifolds

    Bin Shen and Yuhan Zhu

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

  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”, “lower bound”

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

    Xiang-Dong Li

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

  4. 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 “super Perelman”, “Perelman Ricci”, “Harnack type”

  5. Metric Measure Spaces and Synthetic Ricci Bounds – Fundamental Concepts and Recent Developments

    Karl-Theodor Sturm

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

  6. Log gradient estimates for heat type equations on manifolds

    Qi S. Zhang

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

  7. 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 “gradient estimate”, “Harnack type”, “type inequality”

  8. Synthetic notions of Ricci flow for metric measure spaces

    Matthias Erbar, Zhenhao Li and Timo Schultz

    Topics Limit spaces & RCD · Shared terms “metric measure”, “measure spaces”, “time dependent”

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) 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)

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.