Paper · Jul 24, 2021

On the embeddability of the homogeneous Ricci flow and its collapses

Mauro Patrão, Lucas Seco, Llohann D. Sperança

math.DGarXiv:2107.11612
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Abstract

This article grew out of the urge to realize explicit examples of solutions for the Ricci flow as families of isometrically embedded submanifolds, together with its Gromov-Hausdorff collapses. To this aim, we consider the Ricci flow of invariant metrics in a class of flag manifolds. On the one hand, we contrast with a previous result in literature by presenting entire flow lines of invariant metrics realized as orbits of a fixed representation. Indeed, we prove that the subset of realizable metrics has a global attractor with open interior, containing an expressive family of complete flow lines. On the other hand, we prove that certain collapses cannot be realized in any fixed Euclidean space. We provide a detailed picture of the flow, including examples of both realizable and non-realizable flow lines and collapses.

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Submitted
Jul 24, 2021
Subjects
math.DG

Comments: 34 pages, 2 figures

Ricci flowLimit spaces & RCDHomogeneous spaces

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 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 Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces (Francesco Pediconi and Sammy Sbiti, 2021)4Pediconi 2021 On the dynamics of positively curved metrics on SU(3)/T² under the homogeneous Ricci flow (Leonardo F. Cavenaghi, Lino Grama and Ricardo M. Martins, 2023)2Cavenaghi 2023 On the Isometry Group of Immortal Homogeneous Ricci Flows (Roberto Araujo, 2023)6Araujo 2023 Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow (Anusha M. Krishnan, Francesco Pediconi and Sammy Sbiti, 2023)3Krishnan 2023 Long-time behavior of awesome homogeneous Ricci flows (Roberto Araujo, 2023)7Araujo 2023 Riemannian Geometry of G₂-type Real Flag Manifolds (Brian Grajales, Gabriel Rondón and Julieth Saavedra, 2024)5Grajales 2024 Finite extinction time of a family of homogeneous Ricci flows (Roberto Araujo, 2025)8Araujo 2025 On the long time behavior of ancient homogeneous Ricci flows (Anusha M. Krishnan and Francesco Pediconi, 2026)1Krishnan 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 long time behavior of ancient homogeneous Ricci flows

    Anusha M. Krishnan and Francesco Pediconi

    Topics Limit spaces & RCD, Homogeneous spaces · Shared terms “invariant metrics”

  2. On the dynamics of positively curved metrics on under the homogeneous Ricci flow

    Leonardo F. Cavenaghi, Lino Grama and Ricardo M. Martins

    Topics Homogeneous spaces · Shared terms “invariant metrics”, “flag manifold”

  3. Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow

    Anusha M. Krishnan, Francesco Pediconi and Sammy Sbiti

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

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

    Francesco Pediconi and Sammy Sbiti

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

  5. Riemannian Geometry of -type Real Flag Manifolds

    Brian Grajales, Gabriel Rondón and Julieth Saavedra

    Topics Homogeneous spaces · Shared terms “flag manifold”, “representation”

  6. On the Isometry Group of Immortal Homogeneous Ricci Flows

    Roberto Araujo

    Topics Homogeneous spaces

  7. Long-time behavior of awesome homogeneous Ricci flows

    Roberto Araujo

    Topics Homogeneous spaces

  8. Finite extinction time of a family of homogeneous Ricci flows

    Roberto Araujo

    Topics Homogeneous spaces

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) 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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.