Paper · May 7, 2020

Some local Maximum principles along Ricci Flow

Man-Chun Lee, Luen-Fai Tam

math.DGarXiv:2005.03189v2
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Abstract

In this note, we establish a local maximum principle along Ricci flow under scaling invariant curvature condition. This unifies the known preservation of nonnegativity results along Ricci flow with unbounded curvature. By combining with the Dirichlet heat kernel estimates, we also give a more direct proof of Hochard's localized version of a maximum principle given by R. Bamler, E. Cabezas-Rivas, and B. Wilking on the lower bound of curvature conditions.

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Submitted
May 7, 2020
Last revised
Nov 4, 2020 (v2)
Subjects
math.DG

Comments: 25 pages

Ricci flowHeat kernel

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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 2016, plus the canon.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
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    Topics Ricci flow · Same authors Man-Chun Lee, Luen-Fai Tam · Shared terms “principle along”, “maximum principle”

  2. Gaussian upper bounds for the heat kernel on evolving manifolds

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    Topics Heat kernel

  3. Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow

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    Topics Heat kernel · Shared terms “kernel estimates”, “Bamler”

  4. Heat kernel on Ricci shrinkers

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    Topics Heat kernel · Shared terms “maximum principle”

  5. Longtime existence of Kähler Ricci flow and holomorphic sectional curvature

    Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam et al.

    Topics Ricci flow · Same authors Man-Chun Lee, Luen-Fai Tam · Shared terms “unbounded curvature”

  6. Heat kernel on Ricci shrinkers (II)

    Yu Li and Bing Wang

    Topics Heat kernel · Shared terms “Bamler”

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

    Hong Huang

    Topics Heat kernel

  8. Monotonicity of functionals along conformal Ricci flow

    Fengjiang Li, Peng Lu, Jianhong Wang et al.

    Topics Heat kernel

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) Heat kernel The heat kernel of the conjugate heat equation, coupled to Ricci flow, is the probe Perelman and Bamler use to measure geometry. Start with: On the parabolic kernel of the Schrödinger operator (Li et al., 1986)

In the guidebook

  1. 6A.3
    The Heat Equation on ℝⁿBook 6A · The Heat Equation and Its Relatives

    The heat kernel, smoothing, uniqueness, Brownian motion, and a Gaussian that runs to Perelman.

  2. 6A.4
    Fourier Methods and Scale SpaceBook 6A · The Heat Equation and Its Relatives

    Fourier's own method on the circle and the line, smoothing as the decay of high frequencies, and the heat equation as the mathematics of blurring images.

  3. 6A.12
    Entropy, Information and DiffusionBook 6A · The Heat Equation and Its Relatives

    Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.