Paper · Sep 13, 2020

Ancient solutions of the homogeneous Ricci flow on flag manifolds

Stavros Anastassiou, Ioannis Chrysikos

math.DGarXiv:2009.05985v4
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Abstract

For any flag manifold of a compact simple Lie group we describe non-collapsing ancient invariant solutions of the homogeneous unnormalized Ricci flow. Such solutions emerge from an invariant Einstein metric on , and by [BöLS17] they must develop a Type I singularity in their extinction finite time, and also to the past. To illustrate the situation we engage ourselves with the global study of the dynamical system induced by the unnormalized Ricci flow on any flag manifold with second Betti number , for a generic initial invariant metric. We describe the corresponding dynamical systems and present non-collapsed ancient solutions, whose -limit set consists of fixed points at infinity of . Based on the Poincaré compactification method, we show that these fixed points correspond to invariant Einstein metrics and we study their stability properties, illuminating thus the structure of the system's phase space.

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Submitted
Sep 13, 2020
Last revised
Aug 1, 2021 (v4)
Subjects
math.DG, math.DS
Journal
Extracta Mathematicae Vol. 36, Num. 1 (2021), 99-145
DOI
10.17398/2605-5686.36.1.99

Comments: 30 pages, 7 figures

Ricci flowAncient solutionsEinstein metricsSingularities(Non)collapsingHomogeneous spacesStability & uniqueness

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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 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)2Böhm 2016 The canonical Einstein metric on G₂ is dynamically unstable under the Ricci flow (Stuart James Hall, 2018)4Hall 2018 The projected homogeneous Ricci flow and three-isotropy-summands flag manifolds (Lino Grama, Ricardo M. Martins, Mauro Patrão et al., 2020)6Grama 2020 On the Ricci Flow of Homogeneous Metrics on Spheres (Sammy Sbiti, 2020)3Sbiti 2020 On the fundamental group of non-collapsed ancient Ricci flows (Richard H. Bamler, 2021)8Bamler 2021 Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces (Francesco Pediconi and Sammy Sbiti, 2021)5Pediconi 2021 Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow (Anusha M. Krishnan, Francesco Pediconi and Sammy Sbiti, 2023)1Krishnan 2023 Riemannian Geometry of G₂-type Real Flag Manifolds (Brian Grajales, Gabriel Rondón and Julieth Saavedra, 2024)7Grajales 2024
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 · Shared terms “collapsed ancient”

  2. Optimal curvature estimates for homogeneous Ricci flows

    Christoph Böhm, Ramiro Lafuente and Miles Simon

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

  3. On the Ricci Flow of Homogeneous Metrics on Spheres

    Sammy Sbiti

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

  4. The canonical Einstein metric on is dynamically unstable under the Ricci flow

    Stuart James Hall

    Topics Einstein metrics, Homogeneous spaces, Stability & uniqueness · Shared terms “invariant Einstein”, “simple Lie”, “fixed point”

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

    Francesco Pediconi and Sammy Sbiti

    Topics Einstein metrics, Ancient solutions, Homogeneous spaces · Shared terms “collapsed ancient”

  6. The projected homogeneous Ricci flow and three-isotropy-summands flag manifolds

    Lino Grama, Ricardo M. Martins, Mauro Patrão et al.

    Topics (Non)collapsing, Homogeneous spaces · Shared terms “flag manifold”

  7. Riemannian Geometry of -type Real Flag Manifolds

    Brian Grajales, Gabriel Rondón and Julieth Saavedra

    Topics Homogeneous spaces · Shared terms “flag manifold”, “dynamical systems”

  8. On the fundamental group of non-collapsed ancient Ricci flows

    Richard H. Bamler

    Topics (Non)collapsing, Ancient solutions, Singularities · Shared terms “collapsed ancient”

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

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. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

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

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