Paper · Mar 20, 2024

Positive sectional curvature is not preserved under the Ricci flow in dimensions seven and thirteen

David González-Álvaro, Masoumeh Zarei

math.DGarXiv:2403.13764v2
PDF Abstract page

As on arXiv

Abstract

We prove that there exist -invariant metrics on Aloff-Wallach spaces , as well as -invariant metrics on the Berger space , which have positive sectional curvature and evolve under the Ricci flow to metrics with non-positively curved planes.

Comments: Final version

Ricci flowSphere theorems

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 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)4Bettiol 2021 Positive intermediate curvatures and Ricci flow (David González-Álvaro and Masoumeh Zarei, 2023)2González-Álv 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)6Cavenaghi 2023 On the dynamics of a three-dimensional differential system related to the normalized Ricci flow on generalized Wallach spaces (Nurlan Abiev, 2023)5Abiev 2023 On properties of the sets of positively curved Riemannian metrics on generalized Wallach spaces (Nurlan Abiev, 2024)3Abiev 2024 Contraction of hypersurfaces with positive sectional curvature in hyperbolic space (Tianci Luo, Yong Wei and Rong Zhou, 2026)7Luo 2026 Ricci Flow Preserves Positive Sectional Curvature on Homogeneous Spheres (Jason DeVito, David González-Álvaro and Masoumeh Zarei, 2026)1DeVito 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci Flow Preserves Positive Sectional Curvature on Homogeneous Spheres

    Jason DeVito, 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 · Same authors David González-Álvaro, Masoumeh Zarei · Shared terms “positive sectional”

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

    Nurlan Abiev PDF

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

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

    Renato G. Bettiol and Anusha M. Krishnan PDF

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

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

    Nurlan Abiev PDF

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

  6. 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 “positive sectional”, “positively curved”, “Wallach”

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

    Tianci Luo, Yong Wei and Rong ZhouWider flows PDF

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

  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”, “preserved”

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)

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. 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.