Paper · Jan 20, 2023

Heat kernel on Ricci shrinkers (II)

Yu Li, Bing Wang

math.DGarXiv:2301.08430
PDF Abstract page

As on arXiv

Abstract

This paper is the sequel to our study of heat kernels on Ricci shrinkers in. In this paper, we improve many estimates in and extend the recent progress of Bamler. In particular, we drop the compactness and curvature boundedness assumptions and show that the theory of -convergence holds naturally on any Ricci flows induced by Ricci shrinkers.

Comments: 66 pages, 1 figure

Ricci flowRicci solitonsHeat kernelLimit spaces & RCD

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 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 Rigidity of the round cylinders in Ricci shrinkers (Yu Li and Bing Wang, 2021)2Li 2021 Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow (Keita Kunikawa and Yohei Sakurai, 2022)5Kunikawa 2022 Rigidity of complex projective spaces in Ricci shrinkers (Yu Li and Wenjia Zhang, 2022)6Li 2022 Nash Entropy and Heat Kernel on Generalized Ricci Flow (Xilun Li, 2022)8Li 2022 On Kähler Ricci shrinker surfaces (Yu Li and Bing Wang, 2023)3Li 2023 The Ricci-DeTurck flow on complete manifolds (Jing-Bin Cai and Bing Wang, 2026)7Cai 2026 Gromov-Hausdorff convergence of time-slices of singular Ricci flows in dimension three (Yu Li, 2026)4Li 2026 Heat kernel on Ricci shrinker metric measure spaces (Bing Wang and Jie Wang, 2026)1Wang 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 shrinker metric measure spaces

    Bing Wang and Jie Wang PDF

    Topics Heat kernel, Limit spaces & RCD, Ricci solitons · Same author Bing Wang · Shared terms “Ricci shrinkers”, “flow induced”

  2. Rigidity of the round cylinders in Ricci shrinkers

    Yu Li and Bing Wang PDF

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

  3. On Kähler Ricci shrinker surfaces

    Yu Li and Bing Wang PDF

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

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

    Yu Li PDF

    Topics Heat kernel, Limit spaces & RCD · Same author Yu Li

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

    Keita Kunikawa and Yohei Sakurai PDF

    Topics Heat kernel · Shared terms “Bamler”

  6. Rigidity of complex projective spaces in Ricci shrinkers

    Yu Li and Wenjia Zhang PDF

    Topics Limit spaces & RCD, Ricci solitons · Same author Yu Li · Shared terms “Ricci shrinkers”

  7. The Ricci-DeTurck flow on complete manifolds

    Jing-Bin Cai and Bing Wang PDF

    Topics Heat kernel · Same author Bing Wang

  8. Nash Entropy and Heat Kernel on Generalized Ricci Flow

    Xilun Li PDF

    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) 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) 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

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.