Paper · Mar 14, 2020

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

Lino Grama, Ricardo M. Martins, Mauro Patrão, Lucas Seco, Llohann D. Sperança

math.DGarXiv:2004.01511v2
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Abstract

The Ricci flow was introduced by Hamilton and gained its importance through the years. Of special importance is the limiting behavior of the flow and its symmetry properties. Taking this into account, we present a novel normalization for the homogeneous Ricci flow with natural compactness properties. In addition, we present a characterization for Gromov-Hausdorff limits of homogeneous spaces. As a result, we present a detailed picture of the homogeneous Ricci flow for three-isotropy-summands flag manifolds: phase portraits, basins of attractions, conjugation classes and collapsing phenomena. Moreover, we achieve a full classification of the possible Gromov-Hausdorff limits of the aforementioned lines of flow.

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Submitted
Mar 14, 2020
Last revised
Jun 23, 2021 (v2)
Subjects
math.DG, math.DS

Comments: 35 pages, 4 figures. Exposition improved. Minor mistakes corrected

Ricci flow(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 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 Variational aspects of homogeneous geodesics on generalized flag manifolds and applications (Rafaela F. do Prado and Lino Grama, 2016)4Prado 2016 On the stability of harmonic maps under the homogeneous Ricci flow (Rafaela F. do Prado and Lino Grama, 2017)5Prado 2017 Ricci flow on certain homogeneous spaces (Marina Statha, 2020)7Statha 2020 On the embeddability of the homogeneous Ricci flow and its collapses (Mauro Patrão, Lucas Seco and Llohann D. Sperança, 2021)1Patrão 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 The complete dynamics description of positively curved metrics in the Wallach flag manifold SU(3)/T² (Leonardo F. Cavenaghi, Lino Grama, Ricardo M. Martins et al., 2023)3Cavenaghi 2023 Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow (Anusha M. Krishnan, Francesco Pediconi and Sammy Sbiti, 2023)8Krishnan 2023 Homogeneous spaces with two equivalent isotropy summands develop positive Ricci curvature under Ricci flow (Eric Cochran, Arseny Mingajev, Lawrence Mouillé et al., 2026)6Cochran 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 embeddability of the homogeneous Ricci flow and its collapses

    Mauro Patrão, Lucas Seco and Llohann D. Sperança

    Topics Limit spaces & RCD, Homogeneous spaces · Same authors Mauro Patrão, Lucas Seco, Llohann D. Sperança · Shared terms “flag manifold”, “detailed picture”, “Gromov Hausdorff”

  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 · Same authors Lino Grama, Ricardo M. Martins · Shared terms “flag manifold”

  3. The complete dynamics description of positively curved metrics in the Wallach flag manifold

    Leonardo F. Cavenaghi, Lino Grama, Ricardo M. Martins et al.

    Topics Ricci flow · Same authors Lino Grama, Ricardo M. Martins · Shared terms “flag manifold”, “projected”, “addition”

  4. Variational aspects of homogeneous geodesics on generalized flag manifolds and applications

    Rafaela F. do Prado and Lino Grama

    Topics Homogeneous spaces · Same author Lino Grama · Shared terms “flag manifold”

  5. On the stability of harmonic maps under the homogeneous Ricci flow

    Rafaela F. do Prado and Lino Grama

    Topics Homogeneous spaces · Same author Lino Grama

  6. Homogeneous spaces with two equivalent isotropy summands develop positive Ricci curvature under Ricci flow

    Eric Cochran, Arseny Mingajev, Lawrence Mouillé et al.

    Topics Homogeneous spaces · Shared terms “isotropy summands”, “phase portraits”

  7. Ricci flow on certain homogeneous spaces

    Marina Statha

    Topics Homogeneous spaces · Shared terms “isotropy summands”, “flag manifold”

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

    Anusha M. Krishnan, Francesco Pediconi and Sammy Sbiti

    Topics (Non)collapsing, Limit spaces & RCD, Homogeneous spaces · Shared terms “Gromov Hausdorff”

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

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

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.