Paper · Aug 3, 2026

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

Eric Cochran, Arseny Mingajev, Lawrence Mouillé, Nazia Valiyakath

math.DGarXiv:2608.02840
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Abstract

We study normalized Ricci flow on simply connected homogeneous spaces G/H for which the isotropy representation splits into exactly two equivalent irreducible subrepresentations. We prove that every G-invariant metric evolves to one with positive Ricci curvature, and that the family of G-invariant metrics with positive Ricci curvature is forward-invariant under the flow. The proof relies on the fact that the phase portrait of the family of fixed-volume G-invariant metrics can be explicitly visualized.

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Submitted
Aug 3, 2026
Subjects
math.DG

Comments: Comments are welcome!

Ricci flowHomogeneous spacesNumerics & visualization

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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.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Using Ricci Flow in the Construction of Homogeneous Cosmological Models

    Fan Zhang and Lee Lindblom

    Topics Numerics & visualization, Homogeneous spaces

  2. Riemannian Geometry of -type Real Flag Manifolds

    Brian Grajales, Gabriel Rondón and Julieth Saavedra

    Topics Homogeneous spaces · Shared terms “isotropy representation”, “equivalent”

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

    Nurlan A. Abiev

    Topics Homogeneous spaces · Shared terms “normalized Ricci”

  4. 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 “positive Ricci”, “invariant metrics”, “explicitly”

  5. Kähler-Ricci flow on rational homogeneous varieties

    Eder Correa

    Topics Numerics & visualization, Homogeneous spaces · Shared terms “representation”, “explicitly”

  6. Cohomogeneity one 4-dimensional gradient Ricci solitons

    Patrick Donovan

    Topics Numerics & visualization, Homogeneous spaces · Shared terms “simply connected”

  7. Generalized Ricci flow on aligned homogeneous spaces

    Valeria Gutiérrez

    Topics Homogeneous spaces · Shared terms “invariant metrics”

  8. Stability of piecewise flat Ricci flow

    Rory Conboye

    Topics Numerics & visualization

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) Numerics & visualization Computing and visualising Ricci flow: simulations of neckpinches and critical behaviour, pictures of rotationally symmetric flows, and… Start with: Deforming metrics in the direction of their Ricci tensors (DeTurck, 1983)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  2. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

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

  3. 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.