Paper · Oct 27, 2026

An Anisotropic Entropy for the Rotationally Symmetric Ricci Flow

Hongxin Guo, Guozhi Lin

math.DGarXiv:2610.37154
PDF Abstract page

As on arXiv

Abstract

For a rotationally symmetric Ricci flow , , we study the scale-invariant curvature anisotropy introduced by Angenent–Knopf. At the critical power , with , we prove the distributional identity Integration against a conjugate heat kernel gives an anisotropic entropy formula. As an application, we obtain an entropy threshold for Type-I singularities on : if the entropy associated with a singular-time kernel is at most at an earlier regular time, every tangent flow at the corresponding singular point is a round shrinking sphere. This criterion follows from the quantization of the singular-time anisotropic density.

Ricci flowSingularitiesHeat kernelRotational symmetry & warped products

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 Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions (Kin Ming Hui, 2021)8Hui 2021 Hessian estimates for the conjugate heat equation coupled with the Ricci flow (Hong Huang, 2021)2Huang 2021 Tangent Flows of Kähler Metric Flows (Max Hallgren and Wangjian Jian, 2022)4Hallgren 2022 Nash Entropy and Heat Kernel on Generalized Ricci Flow (Xilun Li, 2022)7Li 2022 Heat kernel on Ricci shrinkers (II) (Yu Li and Bing Wang, 2023)3Li 2023 Exponential Asymptotics of Ricci Flow Neckpinch (Hamidreza Mahmoudian, 2023)6Mahmoudian 2023 Rotationally symmetric Ricci Flow on ℝⁿ⁺¹ (Ming Hsiao, 2025)1Hsiao 2025 The Fisher Metric of the Ricci Flow Heat Kernel (Bennett Chow and Robert Koirala, 2026)5Chow 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rotationally symmetric Ricci Flow on

    Ming Hsiao PDF

    Topics Rotational symmetry & warped products, Singularities · Shared terms “symmetric Ricci”, “rotationally symmetric”, “invariant curvature”

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

    Hong Huang PDF

    Topics Heat kernel · Shared terms “conjugate heat”

  3. Heat kernel on Ricci shrinkers (II)

    Yu Li and Bing Wang PDF

    Topics Heat kernel

  4. Tangent Flows of Kähler Metric Flows

    Max Hallgren and Wangjian Jian PDF

    Topics Heat kernel, Singularities · Shared terms “tangent flow”, “conjugate heat”

  5. The Fisher Metric of the Ricci Flow Heat Kernel

    Bennett Chow and Robert Koirala PDF

    Topics Heat kernel · Shared terms “conjugate heat”, “identity”, “criterion”

  6. Exponential Asymptotics of Ricci Flow Neckpinch

    Hamidreza Mahmoudian PDF

    Topics Rotational symmetry & warped products, Singularities · Shared terms “rotationally symmetric”, “Knopf”, “Angenent”

  7. Nash Entropy and Heat Kernel on Generalized Ricci Flow

    Xilun Li PDF

    Topics Heat kernel

  8. Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions

    Kin Ming Hui PDF

    Topics Rotational symmetry & warped products · Shared terms “rotationally symmetric”

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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, 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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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.