Paper · Feb 26, 2020

Ricci flow on certain homogeneous spaces

Marina Statha

math.DGarXiv:2002.12175v2
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Abstract

We study the behavior of the normalized Ricci flow of invariant Riemannian homogeneous metrics at infinity for generalized Wallach spaces, generalized flag manifolds with four isotropy summands and second Betti number equal to one, and the Stiefel manifolds and , with . We use techniques from the theory of differential equations, in particular the Poincaré compactification.

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Submitted
Feb 26, 2020
Last revised
Aug 9, 2020 (v2)
Subjects
math.DG

Comments: 25 pages

Ricci flowHomogeneous 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 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 The projected homogeneous Ricci flow and three-isotropy-summands flag manifolds (Lino Grama, Ricardo M. Martins, Mauro Patrão et al., 2020)3Grama 2020 Ancient solutions of the homogeneous Ricci flow on flag manifolds (Stavros Anastassiou and Ioannis Chrysikos, 2020)6Anastassiou 2020 Long-time behavior of awesome homogeneous Ricci flows (Roberto Araujo, 2023)8Araujo 2023 Riemannian Geometry of G₂-type Real Flag Manifolds (Brian Grajales, Gabriel Rondón and Julieth Saavedra, 2024)5Grajales 2024 Ricci curvature and normalized Ricci flow on generalized Wallach spaces (Nurlan Abiev, 2024)7Abiev 2024 Refined behavior description of the normalized Ricci flow on homogeneous spaces (Nurlan A. Abiev, 2025)1Abiev 2025 Homogeneous spaces with two equivalent isotropy summands develop positive Ricci curvature under Ricci flow (Eric Cochran, Arseny Mingajev, Lawrence Mouillé et al., 2026)2Cochran 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Refined behavior description of the normalized Ricci flow on homogeneous spaces

    Nurlan A. Abiev

    Topics Homogeneous spaces · Shared terms “normalized Ricci”, “generalized Wallach”, “Wallach spaces”

  2. 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”, “normalized Ricci”

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

    Lino Grama, Ricardo M. Martins, Mauro Patrão et al.

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

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

    Rafaela F. do Prado and Lino Grama

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

  5. Riemannian Geometry of -type Real Flag Manifolds

    Brian Grajales, Gabriel Rondón and Julieth Saavedra

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

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

    Stavros Anastassiou and Ioannis Chrysikos

    Topics Homogeneous spaces · Shared terms “flag manifold”, “Poincaré compactification”, “Betti number”

  7. Ricci curvature and normalized Ricci flow on generalized Wallach spaces

    Nurlan Abiev

    Topics Ricci flow · Shared terms “generalized Wallach”, “Wallach spaces”, “normalized Ricci”

  8. Long-time behavior of awesome homogeneous Ricci flows

    Roberto Araujo

    Topics Homogeneous spaces · Shared terms “homogeneous metrics”

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) 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. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  2. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.