Paper · Apr 27, 2021

-invariant Ricci solitons and ancient flows on

Timothy Buttsworth

math.DGarXiv:2104.12996
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Abstract

Consider the standard action of on . We establish the existence of a uniform constant so that any -invariant Ricci soliton on with Einstein constant must have Riemann curvature and volume bounded by , and injectivity radius bounded below by . This observation, coupled with basic numerics, gives strong evidence to suggest that the only -invariant Ricci solitons on are round. We also encounter the so-called `pancake' ancient solution of the Ricci flow.

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Submitted
Apr 27, 2021
Subjects
math.DG

Comments: 20 pages, 4 figures

Ricci flowRicci solitonsAncient solutions

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 2016, 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 On the Ricci iteration for homogeneous metrics on spheres and projective spaces (Timothy Buttsworth, Artem Pulemotov, Yanir A. Rubinstein et al., 2018)3Buttsworth 2018 Complexity of parabolic systems (Tobias Holck Colding and William P. Minicozzi, 2019)7Colding 2019 Eguchi-Hanson singularities in U(2)-invariant Ricci flow (Alexander Appleton, 2019)6Appleton 2019 Classification of ancient flows by sub-affine-critical powers of curvature in ℝ² (Kyeongsu Choi and Liming Sun, 2020)5Choi 2020 Canonical surgeries in rotationally invariant Ricci flow (Timothy Buttsworth, Maximilien Hallgren and Yongjia Zhang, 2022)2Buttsworth 2022 Ancient Ricci flows of bounded girth (Theodora Bourni, Timothy Buttsworth, Ramiro Lafuente et al., 2023)1Bourni 2023 An Ancient Stacked Pancake Solution to Mean Curvature Flow (Alexander Mramor and Louis Yudowitz, 2025)8Mramor 2025 On the long time behavior of ancient homogeneous Ricci flows (Anusha M. Krishnan and Francesco Pediconi, 2026)4Krishnan 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ancient Ricci flows of bounded girth

    Theodora Bourni, Timothy Buttsworth, Ramiro Lafuente et al.

    Topics Ancient solutions · Same author Timothy Buttsworth

  2. Canonical surgeries in rotationally invariant Ricci flow

    Timothy Buttsworth, Maximilien Hallgren and Yongjia Zhang

    Topics Ricci flow · Same author Timothy Buttsworth · Shared terms “invariant Ricci”

  3. On the Ricci iteration for homogeneous metrics on spheres and projective spaces

    Timothy Buttsworth, Artem Pulemotov, Yanir A. Rubinstein et al.

    Topics Ancient solutions · Same author Timothy Buttsworth

  4. On the long time behavior of ancient homogeneous Ricci flows

    Anusha M. Krishnan and Francesco Pediconi

    Topics Ancient solutions, Ricci solitons · Shared terms “injectivity radius”

  5. Classification of ancient flows by sub-affine-critical powers of curvature in

    Kyeongsu Choi and Liming Sun

    Topics Ancient solutions, Ricci solitons

  6. Eguchi-Hanson singularities in U(2)-invariant Ricci flow

    Alexander Appleton

    Topics Ancient solutions · Shared terms “invariant Ricci”

  7. Complexity of parabolic systems

    Tobias Holck Colding and William P. Minicozzi

    Topics Ancient solutions, Ricci solitons

  8. An Ancient Stacked Pancake Solution to Mean Curvature Flow

    Alexander Mramor and Louis YudowitzWider flows

    Topics Ancient solutions

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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

In the guidebook

  1. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  2. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  3. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.