Paper · Jun 16, 2016

Ricci iteration on homogeneous spaces

Artem Pulemotov, Yanir A. Rubinstein

math.DGarXiv:1606.05064v2
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Abstract

The Ricci iteration is a discrete analogue of the Ricci flow. We give the first study of the Ricci iteration on a class of Riemannian manifolds that are not Kähler. The Ricci iteration in the non-Kähler setting exhibits new phenomena. Among them is the existence of so-called ancient Ricci iterations. As we show, these are closely related to ancient Ricci flows and provide the first nontrivial examples of Riemannian metrics to which the Ricci operator can be applied infinitely many times. In some of the cases we study, these ancient Ricci iterations emerge (in the Gromov–Hausdorff topology) from a collapsed Einstein metric and converge smoothly to a second Einstein metric. In the case of compact homogeneous spaces with maximal isotropy, we prove a relative compactness result that excludes collapsing.

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Submitted
Jun 16, 2016
Last revised
Dec 18, 2018 (v2)
Subjects
math.DG
Journal
Transactions of the AMS 371 (2019), pages 6257-6287

Comments: 30 pages

Ricci flowAncient solutionsEinstein metrics(Non)collapsingKähler geometryLimit spaces & RCDHomogeneous spacesLong-time existence & convergence

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 Convergence of the Kähler-Ricci iteration (Tamás Darvas and Yanir A. Rubinstein, 2017)2Darvas 2017 On the Ricci iteration for homogeneous metrics on spheres and projective spaces (Timothy Buttsworth, Artem Pulemotov, Yanir A. Rubinstein et al., 2018)1Buttsworth 2018 On the Ricci Flow of Homogeneous Metrics on Spheres (Sammy Sbiti, 2020)6Sbiti 2020 Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces (Francesco Pediconi and Sammy Sbiti, 2021)3Pediconi 2021 Ricci iterations of well-behaved Kähler metrics (Andrea Loi and Giovanni Placini, 2023)8Loi 2023 Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow (Anusha M. Krishnan, Francesco Pediconi and Sammy Sbiti, 2023)4Krishnan 2023 On the long time behavior of ancient homogeneous Ricci flows (Anusha M. Krishnan and Francesco Pediconi, 2026)7Krishnan 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 · Same authors Artem Pulemotov, Yanir A. Rubinstein · Shared terms “Ricci iteration”

  2. Convergence of the Kähler-Ricci iteration

    Tamás Darvas and Yanir A. Rubinstein

    Topics Einstein metrics, Kähler geometry · Same author Yanir A. Rubinstein · Shared terms “Ricci iteration”, “discrete analogue”

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

    Francesco Pediconi and Sammy Sbiti

    Topics Einstein metrics, Ancient solutions, Limit spaces & RCD · Shared terms “compact homogeneous”, “Hausdorff topology”, “collapsed”

  4. 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 · Shared terms “Hausdorff topology”, “collapsed”, “Gromov”

  5. Optimal curvature estimates for homogeneous Ricci flows

    Christoph Böhm, Ramiro Lafuente and Miles Simon

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

  6. On the Ricci Flow of Homogeneous Metrics on Spheres

    Sammy Sbiti

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

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

    Anusha M. Krishnan and Francesco Pediconi

    Topics Ancient solutions, Limit spaces & RCD, Homogeneous spaces · Shared terms “compact homogeneous”

  8. Ricci iterations of well-behaved Kähler metrics

    Andrea Loi and Giovanni Placini

    Topics Einstein metrics, Kähler geometry · Shared terms “Ricci iteration”, “Ricci operator”

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.