Paper · Dec 3, 2023

Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow

Anusha M. Krishnan, Francesco Pediconi, Sammy Sbiti

math.DGarXiv:2312.01469v3
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Abstract

Collapsed ancient solutions to the homogeneous Ricci flow on compact manifolds occur only on the total space of principal torus bundles. Under an algebraic assumption that guarantees flowing through diagonal metrics and a tameness assumption on the collapsing directions, we prove that such solutions have additional symmetries, i.e., they are invariant under the right action of their collapsing torus. As a byproduct of these additional torus symmetries, we prove that these solutions converge, backward in time, in the Gromov-Hausdorff topology to an Einstein metric on the base of a torus bundle.

Comments: 35 pages, final version, to appear in J. Lond. Math. Soc

Ricci flowAncient solutionsEinstein metrics(Non)collapsingLimit spaces & RCDHomogeneous spaces

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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 2021, 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 embeddability of the homogeneous Ricci flow and its collapses (Mauro Patrão, Lucas Seco and Llohann D. Sperança, 2021)6Patrão 2021 Ricci Flow on Torus Bundles (Dmytro Yeroshkin, 2021)4Yeroshkin 2021 On the fundamental group of non-collapsed ancient Ricci flows (Richard H. Bamler, 2021)3Bamler 2021 Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces (Francesco Pediconi and Sammy Sbiti, 2021)1Pediconi 2021 Collapsed manifolds with local Ricci bounded covering geometry (Xiaochun Rong, 2022)5Rong 2022 On the Isometry Group of Immortal Homogeneous Ricci Flows (Roberto Araujo, 2023)7Araujo 2023 Long-time behavior of awesome homogeneous Ricci flows (Roberto Araujo, 2023)8Araujo 2023 On the long time behavior of ancient homogeneous Ricci flows (Anusha M. Krishnan and Francesco Pediconi, 2026)2Krishnan 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces

    Francesco Pediconi and Sammy Sbiti PDF

    Topics Einstein metrics, Ancient solutions, Limit spaces & RCD · Same authors Francesco Pediconi, Sammy Sbiti · Shared terms “Collapsed ancient”, “Hausdorff topology”, “Gromov Hausdorff”

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

    Anusha M. Krishnan and Francesco Pediconi PDF

    Topics Ancient solutions, Limit spaces & RCD, Homogeneous spaces · Same authors Anusha M. Krishnan, Francesco Pediconi · Shared terms “compact manifolds”

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

    Richard H. Bamler PDF

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

  4. Ricci Flow on Torus Bundles

    Dmytro Yeroshkin PDF

    Topics Einstein metrics, Limit spaces & RCD · Shared terms “torus bundles”

  5. Collapsed manifolds with local Ricci bounded covering geometry

    Xiaochun Rong PDF

    Topics (Non)collapsing, Homogeneous spaces · Shared terms “collapsed”, “torus bundles”, “Gromov”

  6. On the embeddability of the homogeneous Ricci flow and its collapses

    Mauro Patrão, Lucas Seco and Llohann D. Sperança PDF

    Topics Limit spaces & RCD, Homogeneous spaces · Shared terms “Gromov Hausdorff”

  7. On the Isometry Group of Immortal Homogeneous Ricci Flows

    Roberto Araujo PDF

    Topics Homogeneous spaces

  8. Long-time behavior of awesome homogeneous Ricci flows

    Roberto Araujo PDF

    Topics Homogeneous spaces · Shared terms “compact manifolds”

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.