Paper · Oct 9, 2023

The Smale conjecture and min-max theory

Daniel Ketover, Yevgeny Liokumovich

math.DGarXiv:2310.05756v3
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Abstract

We give a new proof of the Smale conjecture for and all lens spaces using minimal surfaces and min-max theory. For , the conjecture was first proved in 2019 by Bamler-Kleiner using Ricci flow.

Comments: To appear in Invent.math

Ricci flowSmale conjecture & diffeomorphismsMinimal & CMC surfaces

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 Uniqueness and stability of Ricci flow through singularities (Richard H. Bamler and Bruce Kleiner, 2017)4Bamler 2017 Rigidity of spherical product Ricci solitons (Ao Sun and Jonathan J. Zhu, 2021)7Sun 2021 The spinorial energy for asymptotically Euclidean Ricci flow (Julius Baldauf and Tristan Ozuch, 2022)5Baldauf 2022 Free boundary minimal disks in convex balls (Robert Haslhofer and Daniel Ketover, 2023)1Haslhofer 2023 The Smale Conjecture and Minimal Legendrian Graph in S²× S³ (Shu-Cheng Chang, Chin-Tung Wu and Liuyang Zhang, 2023)3Chang 2023 Existence of 5 minimal tori in 3-spheres of positive Ricci curvature (Adrian Chun-Pong Chu and Yangyang Li, 2024)2Chu 2024 Navigating string theory field space with geometric flows (Saskia Demulder, Dieter Lust and Thomas Raml, 2024)8Demulder 2024 Differential topology of the spaces of asymptotically stable vector fields and Lyapunov functions (Matthew D. Kvalheim, 2025)6Kvalheim 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Free boundary minimal disks in convex balls

    Robert Haslhofer and Daniel KetoverWider flows PDF

    Topics Minimal & CMC surfaces · Same author Daniel Ketover · Shared terms “max theory”, “min max”

  2. Existence of 5 minimal tori in 3-spheres of positive Ricci curvature

    Adrian Chun-Pong Chu and Yangyang LiWider flows PDF

    Topics Minimal & CMC surfaces · Shared terms “max theory”, “min max”

  3. The Smale Conjecture and Minimal Legendrian Graph in

    Shu-Cheng Chang, Chin-Tung Wu and Liuyang ZhangWider flows PDF

    Topics Smale conjecture & diffeomorphisms

  4. Uniqueness and stability of Ricci flow through singularities

    Richard H. Bamler and Bruce KleinerThe canon PDF

    Topics Smale conjecture & diffeomorphisms

  5. The spinorial energy for asymptotically Euclidean Ricci flow

    Julius Baldauf and Tristan Ozuch PDF

    Topics Minimal & CMC surfaces · Shared terms “min max”

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

    Matthew D. Kvalheim PDF

    Topics Smale conjecture & diffeomorphisms

  7. Rigidity of spherical product Ricci solitons

    Ao Sun and Jonathan J. Zhu PDF

    Topics Smale conjecture & diffeomorphisms

  8. Navigating string theory field space with geometric flows

    Saskia Demulder, Dieter Lust and Thomas Raml PDF

    Topics Smale conjecture & diffeomorphisms

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) 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) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

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

  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.