Paper · May 2, 2017

Remarks on Entropy Formulae for Linear Heat Equation

Yucheng Ji

math.DGarXiv:1705.00765v5
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Abstract

In this note, we prove some new entropy formula for linear heat equation on static Riemannian manifold with nonnegative Ricci curvature. The results are analogies of Cao and Hamilton's entropies for Ricci flow coupled with heat-type equations.

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Submitted
May 2, 2017
Last revised
Jul 20, 2022 (v5)
Subjects
math.DG
Journal
Differential Geometry and its Applications. Volume 84 (2022) 101926
DOI
10.1016/j.difgeo.2022.101926.

Comments: Final version; to appear in Differential Geometry and its Applications

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 W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds (Songzi Li and Xiang-Dong Li, 2017)5Li 2017 Hessian estimates for the conjugate heat equation coupled with the Ricci flow (Hong Huang, 2021)7Huang 2021 Parabolic frequency monotonicity on Ricci flow and Ricci-harmonic flow with bounded curvatures (Chuanhuan Li, Yi Li and Kairui Xu, 2022)6Li 2022 Sharp L² estimates for the drift heat equation on shrinking Ricci solitons (Heather Macbeth, 2024)4Macbeth 2024 A Simple Ricci Flow Proof of the Uniformization Theorem (Yucheng Ji, 2024)3Ji 2024 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)1Li 2024 Log gradient estimates for heat type equations on manifolds (Qi S. Zhang, 2024)2Zhang 2024 Regularizing estimates for positive solutions of the heat equation under geometric flows (Alessandro Goffi and Francesco Pediconi, 2026)8Goffi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 Ricci flow · Shared terms “linear heat”, “heat type”, “type equations”

  2. Log gradient estimates for heat type equations on manifolds

    Qi S. Zhang

    Topics Ricci flow · Shared terms “heat type”, “type equations”, “heat equation”

  3. A Simple Ricci Flow Proof of the Uniformization Theorem

    Yucheng Ji

    Topics Ricci flow · Same author Yucheng Ji · Shared terms “Hamilton”

  4. Sharp estimates for the drift heat equation on shrinking Ricci solitons

    Heather Macbeth

    Topics Ricci flow · Shared terms “heat equation”

  5. -entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds

    Songzi Li and Xiang-Dong Li

    Topics Ricci flow · Shared terms “entropy formula”, “heat equation”, “Riemannian manifold”

  6. Parabolic frequency monotonicity on Ricci flow and Ricci-harmonic flow with bounded curvatures

    Chuanhuan Li, Yi Li and Kairui Xu

    Topics Ricci flow · Shared terms “linear heat”, “heat equation”

  7. Hessian estimates for the conjugate heat equation coupled with the Ricci flow

    Hong Huang

    Topics Ricci flow · Shared terms “heat equation”

  8. Regularizing estimates for positive solutions of the heat equation under geometric flows

    Alessandro Goffi and Francesco Pediconi

    Topics Ricci flow · Shared terms “heat equation”, “Riemannian 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)

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.