Paper · Jan 27, 2022

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

Keita Kunikawa, Yohei Sakurai

math.DGarXiv:2201.11361v3
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Abstract

Bamler-Zhang have developed geometric analysis on Ricci flow with scalar curvature bound. The aim of this paper is to extend their work to various geometric flows. We generalize some of their results to super Ricci flow whose Muller quantity is non-negative, and obtain Gaussian heat kernel estimates.

Comments: 22 pages

Ricci flowHeat kernelScalar curvatureWeak & singular flows

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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 2021, 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 Entropy and heat kernel bounds on a Ricci flow background (Richard H Bamler, 2020)4Bamler 2020 Ricci limit flows and weak solutions (Beomjun Choi and Robert Haslhofer, 2021)7Choi 2021 Nash Entropy and Heat Kernel on Generalized Ricci Flow (Xilun Li, 2022)6Li 2022 Heat kernel on Ricci shrinkers (II) (Yu Li and Bing Wang, 2023)2Li 2023 Almost splitting and quantitative stratification for super Ricci flow (Keita Kunikawa and Yohei Sakurai, 2023)1Kunikawa 2023 Gaussian Volume Functional, Integral Scalar Curvature, and Minimal Super-Ricci Flows (Marco Flaim, Erik Hupp and Karl-Theodor Sturm, 2026)8Flaim 2026 On weak formulations of (super) Ricci flows (Sajjad Lakzian, 2026)5Lakzian 2026 Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits (Lei Zhang and Zhenlei Zhang, 2026)3Zhang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Almost splitting and quantitative stratification for super Ricci flow

    Keita Kunikawa and Yohei Sakurai PDF

    Topics Weak & singular flows, Scalar curvature · Same authors Keita Kunikawa, Yohei Sakurai · Shared terms “super Ricci”, “Muller quantity”, “Bamler”

  2. Heat kernel on Ricci shrinkers (II)

    Yu Li and Bing Wang PDF

    Topics Heat kernel · Shared terms “Bamler”

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

    Lei Zhang and Zhenlei Zhang PDF

    Topics Heat kernel, Scalar curvature · Shared terms “kernel estimates”, “Gaussian”, “Bamler”

  4. Entropy and heat kernel bounds on a Ricci flow background

    Richard H BamlerThe canon PDF

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

  5. On weak formulations of (super) Ricci flows

    Sajjad Lakzian PDF

    Topics Weak & singular flows, Scalar curvature · Shared terms “super Ricci”

  6. Nash Entropy and Heat Kernel on Generalized Ricci Flow

    Xilun Li PDF

    Topics Heat kernel

  7. Ricci limit flows and weak solutions

    Beomjun Choi and Robert Haslhofer PDF

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

  8. Gaussian Volume Functional, Integral Scalar Curvature, and Minimal Super-Ricci Flows

    Marco Flaim, Erik Hupp and Karl-Theodor Sturm PDF

    Topics Weak & singular flows, Scalar curvature · Shared terms “super Ricci”, “Gaussian”

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

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.