The canon · Paper · May 6, 2007

Manifolds with 1/4-pinched Curvature are Space Forms

S. Brendle, R. M. Schoen

math.DGarXiv:0705.0766v3
PDF Abstract page

As on arXiv

Abstract

Brendle–Schoen. The differentiable sphere theorem: Ricci flow beyond dimension three.

Let (M,g_0) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. We show that the Ricci flow deforms g_0 to a constant curvature metric. The proof uses the fact, also established in this paper, that positive isotropic curvature is preserved by the Ricci flow in all dimensions. We also rely on earlier work of Hamilton and of Bohm and Wilking.

Comments: Final version, to appear in J. Amer. Math. Soc

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 Gradient pseudo-Ricci solitons of real hypersurfaces (Mayuko Kon, 2021)8Kon 2021 Manifolds with PIC1 pinched curvature (Man-Chun Lee and Peter M. Topping, 2022)2Lee 2022 Positive intermediate curvatures and Ricci flow (David González-Álvaro and Masoumeh Zarei, 2023)7González-Álv 2023 Classification of compact manifolds with positive isotropic curvature (Hong Huang, 2023)4Huang 2023 Ricci flow and PIC1 (Peter M. Topping, 2023)5Topping 2023 On properties of the sets of positively curved Riemannian metrics on generalized Wallach spaces (Nurlan Abiev, 2024)6Abiev 2024 Positive sectional curvature is not preserved under the Ricci flow in dimensions seven and thirteen (David González-Álvaro and Masoumeh Zarei, 2024)3González-Álv 2024 Ricci flow on manifolds with positive isotropic curvature in all dimensions (Simon Brendle and Raphael Tsiamis, 2026)1Brendle 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 on manifolds with positive isotropic curvature in all dimensions

    Simon Brendle and Raphael Tsiamis PDF

    Topics Sphere theorems · Same author S. Brendle · Shared terms “positive isotropic”, “isotropic curvature”

  2. Manifolds with PIC1 pinched curvature

    Man-Chun Lee and Peter M. Topping PDF

    Topics Sphere theorems · Shared terms “pinched curvature”, “sectional curvature”, “Hamilton”

  3. 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 “sectional curvature”, “preserved”

  4. Classification of compact manifolds with positive isotropic curvature

    Hong Huang PDF

    Topics Sphere theorems · Shared terms “positive isotropic”, “isotropic curvature”, “space forms”

  5. Ricci flow and PIC1

    Peter M. Topping PDF

    Topics Sphere theorems · Shared terms “Riemannian manifolds”

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

    Nurlan Abiev PDF

    Topics Sphere theorems · Shared terms “sectional curvature”

  7. Positive intermediate curvatures and Ricci flow

    David González-Álvaro and Masoumeh Zarei PDF

    Topics Sphere theorems · Shared terms “Böhm”, “Wilking”

  8. Gradient pseudo-Ricci solitons of real hypersurfaces

    Mayuko Kon PDF

    Topics Sphere theorems · Shared terms “space forms”, “Let”

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: Brendle–Schoen: a preserved condition that proves the differentiable sphere theorem.

Foundational paper in the Sphere theorems guide: Brendle–Schoen: manifolds with 1/4-pinched curvature 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.