Paper · Aug 10, 2025

Refined behavior description of the normalized Ricci flow on homogeneous spaces

Nurlan A. Abiev

math.DGarXiv:2508.07391
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Abstract

This article deals with the problems of preserving the Ricci curvature positivity on homogeneous spaces under the normalized Ricci flow (NRF). We found out infinitely many generalized Wallach spaces (GWS) on which the positivity of the Ricci curvature of metrics is preserved when evolved by the NRF. Analogously, the number of GWS is infinite as well, when the positivity of the Ricci curvature can be lost. We also obtain some refinements to our previous results devoted to the case of coincided parameters. A series of examples is discussed.

Comments: 10 figures, 1 table

Ricci flowHomogeneous spaces

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    Nurlan Abiev PDF

    Topics Ricci flow · Same author Nurlan A. Abiev · Shared terms “normalized Ricci”, “generalized Wallach”, “Wallach spaces”

  2. On properties of the sets of positively curved Riemannian metrics on generalized Wallach spaces

    Nurlan Abiev PDF

    Topics Ricci flow · Same author Nurlan A. Abiev · Shared terms “generalized Wallach”, “Wallach spaces”, “normalized Ricci”

  3. On the dynamics of a three-dimensional differential system related to the normalized Ricci flow on generalized Wallach spaces

    Nurlan Abiev PDF

    Topics Ricci flow · Same author Nurlan A. Abiev · Shared terms “normalized Ricci”, “Wallach spaces”, “generalized Wallach”

  4. The Ricci curvature and the normalized Ricci flow on the Stiefel manifolds

    Nurlan Abiev PDF

    Topics Ricci flow · Same author Nurlan A. Abiev · Shared terms “normalized Ricci”, “positivity”

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

    Eric Cochran, Arseny Mingajev, Lawrence Mouillé et al. PDF

    Topics Homogeneous spaces · Shared terms “normalized Ricci”

  6. Long-time behavior of awesome homogeneous Ricci flows

    Roberto Araujo PDF

    Topics Homogeneous spaces

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

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

    Topics Homogeneous spaces · Shared terms “Wallach”

  8. Generalized Ricci flow on aligned homogeneous spaces

    Valeria Gutiérrez PDF

    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) 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.