Paper · Jun 3, 2020

On the Ricci Flow of Homogeneous Metrics on Spheres

Sammy Sbiti

math.DGarXiv:2006.02566
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Abstract

We study the Ricci flow of the four-parameter family of Sp(n+1)-invariant metrics on spheres. We determine their forward behaviour and also classify ancient solutions. In doing so, we exhibit a new one-parameter family of ancient solutions on spheres. These (non-isometric) ancient solutions all have a larger isometry group, namely Sp(n+1)Sp(1), Sp(n+1)U(1), or U(2n+2). Two ancient solutions are non-collapsed and converge, under the backwards flow, to Jensen's second Einstein metric. One solution parametrizes the well known Berger metrics. The rest are new and collapse, under a rescaling of the backwards flow, to Ziller's second homogeneous Einstein metric on complex projective space.

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Submitted
Jun 3, 2020
Subjects
math.DG

Comments: 19 pages, 3 figures

Ricci flowAncient solutionsEinstein metrics(Non)collapsingHomogeneous spaces

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 Optimal curvature estimates for homogeneous Ricci flows (Christoph Böhm, Ramiro Lafuente and Miles Simon, 2016)5Böhm 2016 Ancient Ricci Flow Solutions on Bundles (Peng Lu and Y. K. Wang, 2016)7Lu 2016 Ricci iteration on homogeneous spaces (Artem Pulemotov and Yanir A. Rubinstein, 2016)6Pulemotov 2016 On the Ricci iteration for homogeneous metrics on spheres and projective spaces (Timothy Buttsworth, Artem Pulemotov, Yanir A. Rubinstein et al., 2018)3Buttsworth 2018 Ancient solutions of the homogeneous Ricci flow on flag manifolds (Stavros Anastassiou and Ioannis Chrysikos, 2020)4Anastassiou 2020 Uniqueness of compact ancient solutions to the higher dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al., 2021)8Brendle 2021 Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces (Francesco Pediconi and Sammy Sbiti, 2021)2Pediconi 2021 Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow (Anusha M. Krishnan, Francesco Pediconi and Sammy Sbiti, 2023)1Krishnan 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow

    Anusha M. Krishnan, Francesco Pediconi and Sammy Sbiti

    Topics (Non)collapsing, Einstein metrics, Ancient solutions · Same author Sammy Sbiti · Shared terms “collapsed”, “backward”

  2. Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces

    Francesco Pediconi and Sammy Sbiti

    Topics Einstein metrics, Ancient solutions, Homogeneous spaces · Same author Sammy Sbiti · Shared terms “collapsed”, “rescaling”

  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, Homogeneous spaces · Shared terms “homogeneous metrics”, “projective space”, “complex projective”

  4. Ancient solutions of the homogeneous Ricci flow on flag manifolds

    Stavros Anastassiou and Ioannis Chrysikos

    Topics (Non)collapsing, Einstein metrics, Ancient solutions · Shared terms “collapsed”

  5. Optimal curvature estimates for homogeneous Ricci flows

    Christoph Böhm, Ramiro Lafuente and Miles Simon

    Topics (Non)collapsing, Einstein metrics, Ancient solutions · Shared terms “collapsed”

  6. Ricci iteration on homogeneous spaces

    Artem Pulemotov and Yanir A. Rubinstein

    Topics (Non)collapsing, Einstein metrics, Ancient solutions · Shared terms “collapsed”

  7. Ancient Ricci Flow Solutions on Bundles

    Peng Lu and Y. K. Wang

    Topics (Non)collapsing, Einstein metrics, Ancient solutions · Shared terms “Ziller”, “collapsed”

  8. Uniqueness of compact ancient solutions to the higher dimensional Ricci flow

    Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al.

    Topics (Non)collapsing, 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) 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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

In the guidebook

  1. 12B.4
    Compactness and Universal Estimates for κ-SolutionsBook 12B · κ-Solutions and Surgery

    Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.

  2. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  3. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.