Paper · Jan 17, 2019

Heat kernel on Ricci shrinkers

Yu Li, Bing Wang

math.DGarXiv:1901.05691
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Abstract

In this paper, we systematically study the heat kernel of the Ricci flows induced by Ricci shrinkers. We develop several estimates which are much sharper than their counterparts in general closed Ricci flows. Many classical results, including the optimal Logarithmic Sobolev constant estimate, the Sobolev constant estimate, the no-local-collapsing theorem, the pseudo-locality theorem and the strong maximum principle for curvature tensors, are essentially improved for Ricci flows induced by Ricci shrinkers. Our results provide many necessary tools to analyze short time singularities of the Ricci flows of general dimension.

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Submitted
Jan 17, 2019
Subjects
math.DG

Comments: 91 pages, 4 figures

Ricci flowRicci solitonsSingularities(Non)collapsingHeat kernel

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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 On the structure of Ricci shrinkers (Haozhao Li, Yu Li and Bing Wang, 2018)2Li 2018 The local entropy along Ricci flow—Part B: the pseudo-locality theorems (Bing Wang, 2020)8Wang 2020 Rigidity of the round cylinders in Ricci shrinkers (Yu Li and Bing Wang, 2021)7Li 2021 Heat kernel on Ricci shrinkers (II) (Yu Li and Bing Wang, 2023)1Li 2023 On Kähler Ricci shrinker surfaces (Yu Li and Bing Wang, 2023)4Li 2023 Gromov-Hausdorff convergence of time-slices of singular Ricci flows in dimension three (Yu Li, 2026)6Li 2026 Heat kernel on Ricci shrinker metric measure spaces (Bing Wang and Jie Wang, 2026)3Wang 2026 On the tangent flows of noncollapsed Ricci flow limit spaces (Yu Li and Bing Wang, 2026)5Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Heat kernel on Ricci shrinkers (II)

    Yu Li and Bing Wang

    Topics Heat kernel, Ricci solitons · Same authors Yu Li, Bing Wang · Shared terms “Ricci shrinkers”, “flows induced”

  2. On the structure of Ricci shrinkers

    Haozhao Li, Yu Li and Bing Wang

    Topics (Non)collapsing, Ricci solitons · Same authors Yu Li, Bing Wang · Shared terms “Ricci shrinkers”

  3. Heat kernel on Ricci shrinker metric measure spaces

    Bing Wang and Jie Wang

    Topics Heat kernel, Ricci solitons · Same author Bing Wang · Shared terms “Ricci shrinkers”, “flows induced”, “systematically”

  4. On Kähler Ricci shrinker surfaces

    Yu Li and Bing Wang

    Topics Ricci solitons · Same authors Yu Li, Bing Wang · Shared terms “Ricci shrinkers”

  5. On the tangent flows of noncollapsed Ricci flow limit spaces

    Yu Li and Bing Wang

    Topics (Non)collapsing, Singularities · Same authors Yu Li, Bing Wang

  6. Gromov-Hausdorff convergence of time-slices of singular Ricci flows in dimension three

    Yu Li

    Topics Heat kernel, (Non)collapsing · Same author Yu Li

  7. Rigidity of the round cylinders in Ricci shrinkers

    Yu Li and Bing Wang

    Topics Ricci solitons · Same authors Yu Li, Bing Wang · Shared terms “Ricci shrinkers”

  8. The local entropy along Ricci flow—Part B: the pseudo-locality theorems

    Bing Wang

    Topics (Non)collapsing · Same author Bing Wang · Shared terms “locality theorem”, “pseudo locality”, “collapsing theorem”

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) 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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

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.