The canon · Paper · Dec 17, 2017

Ricci flow and diffeomorphism groups of 3-manifolds

Richard H. Bamler, Bruce Kleiner

math.DGarXiv:1712.06197
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As on arXiv

Abstract

Bamler–Kleiner. Uses Ricci flow through singularities to complete the Generalized Smale Conjecture, apart from RP³.

We complete the proof of the Generalized Smale Conjecture, apart from the case of , and give a new proof of Gabai's theorem for hyperbolic 3-manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms other than and and hyperbolic manifolds, to prove that the moduli space of metrics of constant sectional curvature is contractible. As a corollary, for such a 3-manifold , the inclusion is a homotopy equivalence for any Riemannian metric of constant sectional curvature.

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Submitted
Dec 17, 2017
Subjects
math.DG, math.AP, math.GR, math.GT

Comments: 29 pages, 1 figure

Ricci flowSingularities3-manifoldsSphere theoremsSmale conjecture & diffeomorphismsWeak & singular flows

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)7Brendle 2007 Singular Ricci flows I (Bruce Kleiner and John Lott, 2014)6Kleiner 2014 Uniqueness and stability of Ricci flow through singularities (Richard H. Bamler and Bruce Kleiner, 2017)1Bamler 2017 Rigidity of spherical product Ricci solitons (Ao Sun and Jonathan J. Zhu, 2021)5Sun 2021 The Smale conjecture and min-max theory (Daniel Ketover and Yevgeny Liokumovich, 2023)3Ketover 2023 On the Multiplicity One Conjecture for Mean Curvature Flows of surfaces (Richard H Bamler and Bruce Kleiner, 2023)2Bamler 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 Differential topology of the spaces of asymptotically stable vector fields and Lyapunov functions (Matthew D. Kvalheim, 2025)4Kvalheim 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness and stability of Ricci flow through singularities

    Richard H. Bamler and Bruce KleinerThe canon

    Topics Smale conjecture & diffeomorphisms, 3-manifolds, Singularities · Same authors Richard H. Bamler, Bruce Kleiner · Shared terms “diffeomorphism groups”, “Generalized Smale”

  2. On the Multiplicity One Conjecture for Mean Curvature Flows of surfaces

    Richard H Bamler and Bruce KleinerWider flows

    Topics Weak & singular flows, Singularities · Same authors Richard H. Bamler, Bruce Kleiner

  3. The Smale conjecture and min-max theory

    Daniel Ketover and Yevgeny Liokumovich

    Topics Smale conjecture & diffeomorphisms

  4. Differential topology of the spaces of asymptotically stable vector fields and Lyapunov functions

    Matthew D. Kvalheim

    Topics Smale conjecture & diffeomorphisms · Shared terms “diffeomorphism groups”, “homotopy”, “contractible”

  5. Rigidity of spherical product Ricci solitons

    Ao Sun and Jonathan J. Zhu

    Topics Smale conjecture & diffeomorphisms

  6. Singular Ricci flows I

    Bruce Kleiner and John LottThe canon

    Topics 3-manifolds, Weak & singular flows · Same author Bruce Kleiner

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

    S. Brendle and R. M. SchoenThe canon

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

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

    Nurlan Abiev

    Topics Sphere theorems, 3-manifolds · 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.

Foundational paper in the Smale conjecture & diffeomorphisms guide: Bamler–Kleiner: Ricci flow and diffeomorphism groups of 3-manifolds.

Foundational paper in the 3-manifolds guide: Bamler–Kleiner: Ricci flow and diffeomorphism groups of 3-manifolds.

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… 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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 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.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.