Paper · Jun 19, 2026

Ricci Flow Preserves Positive Sectional Curvature on Homogeneous Spheres

Jason DeVito, David González-Álvaro, Masoumeh Zarei

math.DGarXiv:2606.21417
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Abstract

We prove that the Ricci flow preserves positive sectional curvature on homogeneous spheres and complex projective spaces. In conjunction with prior results, this completes the classification of which homogeneous spaces have positively curved metrics flowing outside the set of positively curved metrics and which do not.

Comments: 44 pages

Ricci flowSphere theoremsHomogeneous 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 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 Manifolds with 1/4-pinched Curvature are Space Forms (S. Brendle and R. M. Schoen, 2007)8Brendle 2007 Ricci flow does not preserve positive sectional curvature in dimension four (Renato G. Bettiol and Anusha M. Krishnan, 2021)3Bettiol 2021 Positive intermediate curvatures and Ricci flow (David González-Álvaro and Masoumeh Zarei, 2023)2González-Álv 2023 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)7Cavenaghi 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)4Cavenaghi 2023 On properties of the sets of positively curved Riemannian metrics on generalized Wallach spaces (Nurlan Abiev, 2024)5Abiev 2024 Positive sectional curvature is not preserved under the Ricci flow in dimensions seven and thirteen (David González-Álvaro and Masoumeh Zarei, 2024)1González-Álv 2024 Contraction of hypersurfaces with positive sectional curvature in hyperbolic space (Tianci Luo, Yong Wei and Rong Zhou, 2026)6Luo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Positive sectional curvature is not preserved under the Ricci flow in dimensions seven and thirteen

    David González-Álvaro and Masoumeh Zarei PDF

    Topics Sphere theorems · Same authors David González-Álvaro, Masoumeh Zarei · Shared terms “positive sectional”, “sectional curvature”, “positively curved”

  2. Positive intermediate curvatures and Ricci flow

    David González-Álvaro and Masoumeh Zarei PDF

    Topics Sphere theorems, Homogeneous spaces · Same authors David González-Álvaro, Masoumeh Zarei · Shared terms “positive sectional”, “preserves”

  3. Ricci flow does not preserve positive sectional curvature in dimension four

    Renato G. Bettiol and Anusha M. Krishnan PDF

    Topics Sphere theorems, Homogeneous spaces · Shared terms “preserves positive”, “positive sectional”, “sectional curvature”

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

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

    Topics Sphere theorems · Shared terms “curved metrics”, “positively curved”, “positive sectional”

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

    Nurlan Abiev PDF

    Topics Sphere theorems · Shared terms “positively curved”, “positive sectional”, “sectional curvature”

  6. Contraction of hypersurfaces with positive sectional curvature in hyperbolic space

    Tianci Luo, Yong Wei and Rong ZhouWider flows PDF

    Topics Sphere theorems, Homogeneous spaces · Shared terms “positive sectional”, “sectional curvature”

  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 “curved metrics”, “positively curved”

  8. Manifolds with 1/4-pinched Curvature are Space Forms

    S. Brendle and R. M. SchoenThe canon PDF

    Topics Sphere theorems · Shared terms “sectional curvature”

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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988) 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.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  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.