Paper · Sep 18, 2019

Ricci flow and contractibility of spaces of metrics

Richard H. Bamler, Bruce Kleiner

math.DGarXiv:1909.08710
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Abstract

We show that the space of metrics of positive scalar curvature on any 3-manifold is either empty or contractible. Second, we show that the diffeomorphism group of every 3-dimensional spherical space form deformation retracts to its isometry group. This proves the Generalized Smale Conjecture. Our argument is independent of Hatcher's theorem in the case and in particular it gives a new proof of the case.

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Submitted
Sep 18, 2019
Subjects
math.DG, math.AP, math.GT

Comments: 111 pages

Ricci flow3-manifoldsSphere theoremsSmale conjecture & diffeomorphismsScalar curvature

Its neighbourhood

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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 Uniqueness and stability of Ricci flow through singularities (Richard H. Bamler and Bruce Kleiner, 2017)2Bamler 2017 Ricci flow and diffeomorphism groups of 3-manifolds (Richard H. Bamler and Bruce Kleiner, 2017)1Bamler 2017 Constrained deformations of positive scalar curvature metrics (Alessandro Carlotto and Chao Li, 2019)6Carlotto 2019 On the rotational symmetry of 3-dimensional κ-solutions (Richard H. Bamler and Bruce Kleiner, 2019)3Bamler 2019 Four-Dimensional Steady Gradient Ricci Solitons with 3-Cylindrical Tangent Flows at Infinity (Richard Bamler, Bennett Chow, Yuxing Deng et al., 2021)5Bamler 2021 The Smale conjecture and min-max theory (Daniel Ketover and Yevgeny Liokumovich, 2023)4Ketover 2023 On the Multiplicity One Conjecture for Mean Curvature Flows of surfaces (Richard H Bamler and Bruce Kleiner, 2023)8Bamler 2023 Differential topology of the spaces of asymptotically stable vector fields and Lyapunov functions (Matthew D. Kvalheim, 2025)7Kvalheim 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 diffeomorphism groups of 3-manifolds

    Richard H. Bamler and Bruce Kleiner

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

  2. Uniqueness and stability of Ricci flow through singularities

    Richard H. Bamler and Bruce Kleiner

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

  3. On the rotational symmetry of 3-dimensional -solutions

    Richard H. Bamler and Bruce Kleiner

    Topics 3-manifolds · Same authors Richard H. Bamler, Bruce Kleiner · Shared terms “independent”

  4. The Smale conjecture and min-max theory

    Daniel Ketover and Yevgeny Liokumovich

    Topics Smale conjecture & diffeomorphisms

  5. Four-Dimensional Steady Gradient Ricci Solitons with -Cylindrical Tangent Flows at Infinity

    Richard Bamler, Bennett Chow, Yuxing Deng et al.

    Topics Sphere theorems · Same author Richard H. Bamler · Shared terms “dimensional spherical”, “spherical space”, “space form”

  6. Constrained deformations of positive scalar curvature metrics

    Alessandro Carlotto and Chao Li

    Topics Smale conjecture & diffeomorphisms, 3-manifolds, Scalar curvature · Shared terms “positive scalar”

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

    Matthew D. Kvalheim

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

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

    Richard H Bamler and Bruce KleinerWider flows

    Same authors Richard H. Bamler, Bruce Kleiner · Shared terms “spherical”

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: contractibility of spaces of metrics.

Foundational paper in the Scalar curvature guide: Bamler–Kleiner: contractibility of spaces of PSC metrics on 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) 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) Smale conjecture & diffeomorphisms Once Ricci flow through singularities is unique and continuous, it can deform whole families of metrics at once. Start with: A Proof of the Smale Conjecture, Diff(S 3 ) ≃O(4) (Hatcher, 1983)

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. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.