Paper · Oct 12, 2021

Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces

Francesco Pediconi, Sammy Sbiti

math.DGarXiv:2110.05818v2
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Abstract

We prove a general existence theorem for collapsed ancient solutions to the Ricci flow on compact homogeneous spaces and we show that they converge in the Gromov-Hausdorff topology, under a suitable rescaling, to an Einstein metric on the base of a torus fibration. This construction generalizes all previous known examples in the literature.

Comments: 15 pages; Theorem A has been improved by removing the G-non-degeneracy hypothesis; Comments are welcome

Ricci flowAncient solutionsEinstein metricsLimit spaces & RCDHomogeneous spaces

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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 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)4Patrão 2021 On the fundamental group of non-collapsed ancient Ricci flows (Richard H. Bamler, 2021)3Bamler 2021 Bismut Ricci flat manifolds with symmetries (Fabio Podestà and Alberto Raffero, 2022)6Podestà 2022 Bismut Ricci flat generalized metrics on compact homogeneous spaces (including a Corrigendum) (Jorge Lauret and Cynthia E. Will, 2023)8Lauret 2023 On the rigidity of Ricci shrinkers (Yu Li and Wenjia Zhang, 2023)7Li 2023 Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow (Anusha M. Krishnan, Francesco Pediconi and Sammy Sbiti, 2023)1Krishnan 2023 Refined behavior description of the normalized Ricci flow on homogeneous spaces (Nurlan A. Abiev, 2025)5Abiev 2025 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. Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow

    Anusha M. Krishnan, 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 author Francesco Pediconi · Shared terms “compact homogeneous”

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

    Richard H. Bamler PDF

    Topics Ancient solutions · Shared terms “Collapsed ancient”

  4. 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”, “literature”

  5. Refined behavior description of the normalized Ricci flow on homogeneous spaces

    Nurlan A. Abiev PDF

    Topics Homogeneous spaces

  6. Bismut Ricci flat manifolds with symmetries

    Fabio Podestà and Alberto Raffero PDF

    Topics Homogeneous spaces · Shared terms “compact homogeneous”

  7. On the rigidity of Ricci shrinkers

    Yu Li and Wenjia Zhang PDF

    Topics Einstein metrics, Limit spaces & RCD · Shared terms “Hausdorff topology”, “Gromov Hausdorff”

  8. Bismut Ricci flat generalized metrics on compact homogeneous spaces (including a Corrigendum)

    Jorge Lauret and Cynthia E. Will PDF

    Topics Homogeneous spaces · Shared terms “compact homogeneous”, “Einstein”

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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

  1. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  2. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.

  3. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.