Paper · Dec 18, 2024

On some new Ricci flow invariant curvature conditions

Yiyan Xu

math.DGarXiv:2412.13633
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As on arXiv

Abstract

We would like to study new Ricci flow invariant curvature conditions. Specifically, we provide quantitative evidence for an unpublished conjecture of Böhm and Wilking. As an application, we study the topology of manifolds with pinched curvature.

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Submitted
Dec 18, 2024
Subjects
math.DG
Ricci flow

Its neighbourhood

Connected papers

The 7 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 7 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 Manifolds with 1/4-pinched Curvature are Space Forms (S. Brendle and R. M. Schoen, 2007)1Brendle 2007 Three-manifolds with non-negatively pinched Ricci curvature (Man-Chun Lee and Peter M. Topping, 2022)6Lee 2022 Manifolds with PIC1 pinched curvature (Man-Chun Lee and Peter M. Topping, 2022)5Lee 2022 Positive intermediate curvatures and Ricci flow (David González-Álvaro and Masoumeh Zarei, 2023)3González-Álv 2023 Long-time behavior of awesome homogeneous Ricci flows (Roberto Araujo, 2023)4Araujo 2023 On an invariant curvature cone along 4-dimensional Ricci flow (Hongting Ding, Shaochuang Huang and Zhuo Peng, 2026)2Ding 2026 Characterization of Sasakian Manifolds by Curvature Conditions (Sourav Nayak, Vladimir Rovenski and Dhriti Sundar Patra, 2026)7Nayak 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Manifolds with 1/4-pinched Curvature are Space Forms

    S. Brendle and R. M. SchoenThe canon

    Topics Ricci flow · Shared terms “pinched curvature”, “Böhm”, “Wilking”

  2. On an invariant curvature cone along 4-dimensional Ricci flow

    Hongting Ding, Shaochuang Huang and Zhuo Peng

    Topics Ricci flow · Shared terms “invariant curvature”

  3. Positive intermediate curvatures and Ricci flow

    David González-Álvaro and Masoumeh Zarei

    Topics Ricci flow · Shared terms “curvature condition”, “Böhm”, “Wilking”

  4. Long-time behavior of awesome homogeneous Ricci flows

    Roberto Araujo

    Topics Ricci flow · Shared terms “flow invariant”

  5. Manifolds with PIC1 pinched curvature

    Man-Chun Lee and Peter M. Topping

    Topics Ricci flow · Shared terms “pinched curvature”

  6. Three-manifolds with non-negatively pinched Ricci curvature

    Man-Chun Lee and Peter M. Topping

    Topics Ricci flow · Shared terms “invariant curvature”

  7. Characterization of Sasakian Manifolds by Curvature Conditions

    Sourav Nayak, Vladimir Rovenski and Dhriti Sundar Patra

    Topics Ricci flow · Shared terms “curvature condition”

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)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.