The canon · Paper · Jun 8, 2006

Manifolds with positive curvature operators are space forms

Christoph Boehm, Burkhard Wilking

math.DGarXiv:math/0606187
View PDF View HTML arXiv page

As on arXiv

Abstract

Böhm–Wilking. Confirms a conjecture of Hamilton: in every dimension, positive curvature operators flow to constant curvature.

We confirm a conjecture of Hamilton: On compact manifolds the normalized Ricci flow evolves metrics with positive curvature operators to limit metrics with constant curvature.

View PDF View HTML arXiv page
Submitted
Jun 8, 2006
Subjects
math.DG, math.AP
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)2Brendle 2007 Ancient Ricci flows of bounded girth (Theodora Bourni, Timothy Buttsworth, Ramiro Lafuente et al., 2023)5Bourni 2023 Ricci flow and PIC1 (Peter M. Topping, 2023)1Topping 2023 On the dynamics of a three-dimensional differential system related to the normalized Ricci flow on generalized Wallach spaces (Nurlan Abiev, 2023)7Abiev 2023 Rigidity of area non-increasing maps (Man-Chun Lee, Luen-Fai Tam and Jingbo Wan, 2023)8Lee 2023 On properties of the sets of positively curved Riemannian metrics on generalized Wallach spaces (Nurlan Abiev, 2024)4Abiev 2024 Positive sectional curvature is not preserved under the Ricci flow in dimensions seven and thirteen (David González-Álvaro and Masoumeh Zarei, 2024)6González-Álv 2024 Steady Gradient Ricci Solitons with O(p)× O(q) Symmetry (Lucas Lavoyer and Luke T. Peachey, 2025)3Lavoyer 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci flow and PIC1

    Peter M. Topping

    Topics Sphere theorems · Shared terms “positive curvature”

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

    S. Brendle and R. M. SchoenThe canon

    Topics Sphere theorems · Shared terms “space forms”, “constant curvature”, “Hamilton”

  3. Steady Gradient Ricci Solitons with Symmetry

    Lucas Lavoyer and Luke T. Peachey

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

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

    Nurlan Abiev

    Topics Sphere theorems · Shared terms “normalized Ricci”

  5. Ancient Ricci flows of bounded girth

    Theodora Bourni, Timothy Buttsworth, Ramiro Lafuente et al.

    Topics Sphere theorems · Shared terms “positive curvature”, “curvature operator”, “Hamilton”

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

    David González-Álvaro and Masoumeh Zarei

    Topics Sphere theorems

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

    Nurlan Abiev

    Topics Sphere theorems · Shared terms “normalized Ricci”, “flow evolves”

  8. Rigidity of area non-increasing maps

    Man-Chun Lee, Luen-Fai Tam and Jingbo Wan

    Topics Sphere theorems · Shared terms “positive 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.

Foundational paper in the Curvature pinching guide: Böhm–Wilking: positive curvature operators are space forms, via new invariant cones.

Foundational paper in the Ricci flow guide: Böhm–Wilking: invariant cones and convergence for positive curvature operator in all dimensions.

Foundational paper in the Sphere theorems guide: Böhm–Wilking: manifolds with positive curvature operators are space forms.

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.