Paper · Jul 20, 2020

The Entropy of Ricci Flows with Type-I Scalar Curvature Bounds

Max Hallgren

math.DGarXiv:2007.10376v3
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Abstract

In this paper, we extend the theory of Ricci flows satisfying a Type-I scalar curvature condition at a finite-time singularity. In [Bam16], Bamler showed that a Type-I rescaling procedure will produce a singular shrinking gradient Ricci soliton with singularities of codimension 4. We prove that the entropy of a conjugate heat kernel based at the singular time converges to the soliton entropy of the singular soliton, and use this to characterize the singular set of the Ricci flow solution in terms of a heat kernel density function. This generalizes results previously only known with the stronger assumption of a Type-I curvature bound. We also show that in dimension 4, the singular Ricci soliton is smooth away from finitely many points, which are conical smooth orbifold singularities

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Submitted
Jul 20, 2020
Last revised
Feb 28, 2022 (v3)
Subjects
math.DG

Comments: Fixed several typos and small mistakes

Ricci flowRicci solitonsSingularitiesHeat kernel4-manifoldsScalar curvatureWeak & singular flows

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 Gaussian upper bounds for the heat kernel on evolving manifolds (Reto Buzano and Louis Yudowitz, 2020)8Buzano 2020 The sharp Li-Yau equality on Shrinking Ricci Solitons with applications (Jason Ledwidge, 2020)7Ledwidge 2020 Ricci Flow with Ricci Curvature and Volume Bounded Below (Max Hallgren, 2021)6Hallgren 2021 Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow (Keita Kunikawa and Yohei Sakurai, 2022)1Kunikawa 2022 Tangent Flows of Kähler Metric Flows (Max Hallgren and Wangjian Jian, 2022)2Hallgren 2022 Heat kernel on Ricci shrinkers (II) (Yu Li and Bing Wang, 2023)4Li 2023 Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits (Lei Zhang and Zhenlei Zhang, 2026)3Zhang 2026 An Anisotropic Entropy for the Rotationally Symmetric Ricci Flow (Hongxin Guo and Guozhi Lin, 2026)5Guo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow

    Keita Kunikawa and Yohei Sakurai

    Topics Heat kernel, Weak & singular flows, Scalar curvature · Shared terms “curvature bounds”, “Bamler”

  2. Tangent Flows of Kähler Metric Flows

    Max Hallgren and Wangjian Jian

    Topics Heat kernel, Singularities · Same author Max Hallgren · Shared terms “conjugate heat”

  3. Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits

    Lei Zhang and Zhenlei Zhang

    Topics Heat kernel, Scalar curvature, Singularities · Shared terms “conjugate heat”, “singular shrinking”, “singular time”

  4. Heat kernel on Ricci shrinkers (II)

    Yu Li and Bing Wang

    Topics Heat kernel, Ricci solitons · Shared terms “Bamler”

  5. An Anisotropic Entropy for the Rotationally Symmetric Ricci Flow

    Hongxin Guo and Guozhi Lin

    Topics Heat kernel, Singularities · Shared terms “singular time”, “conjugate heat”

  6. Ricci Flow with Ricci Curvature and Volume Bounded Below

    Max Hallgren

    Topics 4-manifolds, Singularities · Same author Max Hallgren · Shared terms “orbifold”, “time singularity”, “finite time”

  7. The sharp Li-Yau equality on Shrinking Ricci Solitons with applications

    Jason Ledwidge

    Topics Heat kernel, 4-manifolds, Singularities · Shared terms “conjugate heat”

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

    Reto Buzano and Louis Yudowitz

    Topics Heat kernel · Shared terms “previously”

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) 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.