Paper · Jul 30, 2021

Nontrivial breathers for Ricci flow

Peter M. Topping

math.DGarXiv:2107.14676v2
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Abstract

Perelman has proved that there cannot exist a nontrivial breather for Ricci flow on a closed manifold. Here we construct nontrivial expanding breathers for Ricci flow in all dimensions when the underlying manifold is allowed to be noncompact.

Comments: Version 2 contains new remark pointing out that the same technique works for other geometric flows such as mean curvature flow

Ricci flowBoundaries & non-compact

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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 No breather theorems for the mean curvature flow (Liang Cheng and Yongjia Zhang, 2021)1Cheng 2021 Time zero regularity of Ricci flow (Man-Chun Lee and Peter M. Topping, 2022)7Lee 2022 Three-manifolds with non-negatively pinched Ricci curvature (Man-Chun Lee and Peter M. Topping, 2022)4Lee 2022 Manifolds with PIC1 pinched curvature (Man-Chun Lee and Peter M. Topping, 2022)3Lee 2022 All two-dimensional expanding Ricci solitons (Luke T. Peachey and Peter M. Topping, 2023)5Peachey 2023 Ricci flow and PIC1 (Peter M. Topping, 2023)8Topping 2023 PIC1 pinched manifolds are flat or compact (Alix Deruelle, Man-Chun Lee, Felix Schulze et al., 2026)6Deruelle 2026 Asymptotic Profiles and Non-Trivial Breathers in Kahler-Ricci Flow (Longteng Chen, 2026)2Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. No breather theorems for the mean curvature flow

    Liang Cheng and Yongjia ZhangWider flows PDF

    Topics Boundaries & non-compact · Shared terms “breather”

  2. Asymptotic Profiles and Non-Trivial Breathers in Kahler-Ricci Flow

    Longteng Chen PDF

    Topics Boundaries & non-compact · Shared terms “breather”

  3. Manifolds with PIC1 pinched curvature

    Man-Chun Lee and Peter M. Topping PDF

    Topics Boundaries & non-compact · Same author Peter M. Topping

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

    Man-Chun Lee and Peter M. Topping PDF

    Topics Boundaries & non-compact · Same author Peter M. Topping

  5. All two-dimensional expanding Ricci solitons

    Luke T. Peachey and Peter M. Topping PDF

    Topics Ricci flow · Same author Peter M. Topping · Shared terms “nontrivial”

  6. PIC1 pinched manifolds are flat or compact

    Alix Deruelle, Man-Chun Lee, Felix Schulze et al. PDF

    Topics Boundaries & non-compact · Same author Peter M. Topping

  7. Time zero regularity of Ricci flow

    Man-Chun Lee and Peter M. Topping PDF

    Topics Ricci flow · Same author Peter M. Topping

  8. Ricci flow and PIC1

    Peter M. Topping PDF

    Topics Ricci flow · Same author Peter M. Topping

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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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.