Paper · Jan 4, 2020

Gradient estimate via two-point function under Ricci flow

Min Chen

math.DGarXiv:2001.01024
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As on arXiv

Abstract

We derive estimates relating the values of a solution at any two points to the distance between the points, for quasilinear parabolic equations on compact Riemannian manifolds under the Ricci flow.

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Submitted
Jan 4, 2020
Subjects
math.DG
Ricci flow

Its neighbourhood

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 Elliptic gradient estimates for a nonlinear heat equation and applications (Jia-Yong Wu, 2016)5Wu 2016 Some new gradient estimates for two nonlinear parabolic equations under Ricci flow (Wen Wang and Hui Zhou, 2017)1Wang 2017 Matrix Inequality for the Laplace Equation (Jiewon Park, 2017)8Park 2017 Gradient estimates and Liouville type theorems for Poisson equations (Nguyen Thac Dung and Nguyen Ngoc Khanh, 2018)4Dung 2018 Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with evolving metrics and potentials (Ali Taheri and Vahideh Vahidifar, 2023)3Taheri 2023 Log gradient estimates for heat type equations on manifolds (Qi S. Zhang, 2024)2Zhang 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 Gradient estimates for a parabolic partial differential equation under the Ricci-Bourguignon flow (José N. V. Gomes, Willian I. Tokura and Hikaru Yamamoto, 2026)7Gomes 2026
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 Ricci flow · Shared terms “gradient estimates”, “parabolic equations”

  2. Log gradient estimates for heat type equations on manifolds

    Qi S. Zhang

    Topics Ricci flow · Shared terms “gradient estimates”

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

    Ali Taheri and Vahideh Vahidifar

    Topics Ricci flow · Shared terms “gradient estimates”, “parabolic equations”

  4. Gradient estimates and Liouville type theorems for Poisson equations

    Nguyen Thac Dung and Nguyen Ngoc Khanh

    Shared terms “gradient estimates”, “parabolic equations”

  5. Elliptic gradient estimates for a nonlinear heat equation and applications

    Jia-Yong Wu

    Shared terms “gradient estimates”

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

    Juanling Lu and Yu Zheng

    Topics Ricci flow · Shared terms “gradient estimates”

  7. Gradient estimates for a parabolic partial differential equation under the Ricci-Bourguignon flow

    José N. V. Gomes, Willian I. Tokura and Hikaru Yamamoto

    Topics Ricci flow · Shared terms “gradient estimates”

  8. Matrix Inequality for the Laplace Equation

    Jiewon Park

    Topics Ricci flow · Shared terms “gradient estimates”

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)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.