Paper · Apr 10, 2016

Optimal curvature estimates for homogeneous Ricci flows

Christoph Böhm, Ramiro Lafuente, Miles Simon

math.DGarXiv:1604.02625v2
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Abstract

We prove uniform curvature estimates for homogeneous Ricci flows: For a solution defined on the norm of the curvature tensor at time is bounded by the maximum of and . This is used to show that solutions with finite extinction time are Type I, immortal solutions are Type III and ancient solutions are Type I, where all the constants involved depend only on the dimension . A further consequence is that a non-collapsed homogeneous ancient solution on a compact homogeneous space emerges from a unique Einstein metric on the same space. The above curvature estimates are proved using a gap theorem for Ricci-flatness on homogeneous spaces. The proof of this gap theorem is by contradiction and uses a local convergence result, which holds without symmetry assumptions.

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Submitted
Apr 10, 2016
Last revised
Jun 1, 2016 (v2)
Subjects
math.DG

Comments: Fixed up the formulation of the weak convergence theorem in Section 2

Ricci flowAncient solutionsEinstein metricsRicci-flat & Calabi–YauSingularities(Non)collapsingHomogeneous 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 Ricci iteration on homogeneous spaces (Artem Pulemotov and Yanir A. Rubinstein, 2016)7Pulemotov 2016 Immortal homogeneous Ricci flows (Christoph Böhm and Ramiro A. Lafuente, 2017)1Böhm 2017 The Ricci flow on solvmanifolds of real type (Christoph Böhm and Ramiro A. Lafuente, 2017)2Böhm 2017 On the Ricci Flow of Homogeneous Metrics on Spheres (Sammy Sbiti, 2020)8Sbiti 2020 Ancient solutions of the homogeneous Ricci flow on flag manifolds (Stavros Anastassiou and Ioannis Chrysikos, 2020)5Anastassiou 2020 Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces (Francesco Pediconi and Sammy Sbiti, 2021)3Pediconi 2021 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)6Krishnan 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Immortal homogeneous Ricci flows

    Christoph Böhm and Ramiro A. Lafuente

    Topics Homogeneous spaces, Long-time existence & convergence · Same authors Christoph Böhm, Ramiro Lafuente · Shared terms “curvature estimates”, “immortal”

  2. The Ricci flow on solvmanifolds of real type

    Christoph Böhm and Ramiro A. Lafuente

    Topics Homogeneous spaces · Same authors Christoph Böhm, Ramiro Lafuente

  3. 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 “compact homogeneous”, “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 “collapsed”

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

    Stavros Anastassiou and Ioannis Chrysikos

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

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

    Anusha M. Krishnan and Francesco Pediconi

    Topics Ancient solutions, Homogeneous spaces, Long-time existence & convergence · Shared terms “compact homogeneous”, “curvature tensor”

  7. Ricci iteration on homogeneous spaces

    Artem Pulemotov and Yanir A. Rubinstein

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

  8. On the Ricci Flow of Homogeneous Metrics on Spheres

    Sammy Sbiti

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

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.

Foundational paper in the Homogeneous spaces guide: Böhm–Lafuente–Simon: optimal curvature estimates for homogeneous Ricci flows.

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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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.