Paper · Dec 25, 2021

Ricci flow does not preserve positive sectional curvature in dimension four

Renato G. Bettiol, Anusha M. Krishnan

math.DGarXiv:2112.13291Calc. Var. Partial Differential Equations 62 (2023), no. 1, Paper No. 13
PDF Abstract page

As on arXiv

Abstract

We find examples of cohomogeneity one metrics on and with positive sectional curvature that lose this property when evolved via Ricci flow. These metrics are arbitrarily small perturbations of Grove–Ziller metrics with flat planes that become instantly negatively curved under Ricci flow.

Comments: LaTeX2e, 18 pages, 2 figures

Ricci flow4-manifoldsSphere theoremsHomogeneous 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 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 Positive intermediate curvatures and Ricci flow (David González-Álvaro and Masoumeh Zarei, 2023)4González-Álv 2023 On the dynamics of a three-dimensional differential system related to the normalized Ricci flow on generalized Wallach spaces (Nurlan Abiev, 2023)8Abiev 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)2González-Álv 2024 Geometry and Analysis of Gradient Ricci Solitons in Dimension Four (Xiaodong Cao and Hung Tran, 2024)6Cao 2024 Geometry and Topology of Gradient Shrinking Sasaki-Ricci Solitons (Shu-Cheng Chang, Yingbo Han and Chin-Tung Wu, 2025)7Chang 2025 Contraction of hypersurfaces with positive sectional curvature in hyperbolic space (Tianci Luo, Yong Wei and Rong Zhou, 2026)3Luo 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, Homogeneous spaces · Shared terms “preserves positive”, “positive sectional”, “sectional curvature”

  2. 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 · Shared terms “positive sectional”, “sectional curvature”

  3. 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”

  4. Positive intermediate curvatures and Ricci flow

    David González-Álvaro and Masoumeh Zarei PDF

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

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

    Nurlan Abiev PDF

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

  6. Geometry and Analysis of Gradient Ricci Solitons in Dimension Four

    Xiaodong Cao and Hung Tran PDF

    Topics 4-manifolds · Shared terms “dimension four”

  7. Geometry and Topology of Gradient Shrinking Sasaki-Ricci Solitons

    Shu-Cheng Chang, Yingbo Han and Chin-Tung Wu PDF

    Topics Sphere theorems, 4-manifolds · Shared terms “positive sectional”, “dimension four”, “sectional curvature”

  8. 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 “positive sectional”, “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) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 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.