The canon · Paper · Sep 13, 2017

Uniqueness and stability of Ricci flow through singularities

Richard H. Bamler, Bruce Kleiner

math.DGarXiv:1709.04122v2
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Abstract

Bamler–Kleiner. Singular Ricci flows are unique and stable.

We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3-manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies Perelman's conjecture. We also show that this flow through singularities depends continuously on its initial condition and that it may be obtained as a limit of Ricci flows with surgery. Our results have applications to the study of diffeomorphism groups of three manifolds — in particular to the Generalized Smale Conjecture — which will appear in a subsequent paper.

Comments: 182 pages, 10 figures, minor corrections

Ricci flowSingularitiesSurgery3-manifoldsSmale conjecture & diffeomorphismsStability & uniqueness

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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 Notes on Perelman's papers (Bruce Kleiner and John Lott, 2006)1Kleiner 2006 Singular Ricci flows I (Bruce Kleiner and John Lott, 2014)2Kleiner 2014 Structure theory of non-collapsed limits of Ricci flows (Richard H Bamler, 2020)5Bamler 2020 Cobordism, Singularities and the Ricci Flow Conjecture (David Martín Velázquez, Davide De Biasio and Dieter Lust, 2022)7Velázquez 2022 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)4Bamler 2023 The Smale Conjecture and Minimal Legendrian Graph in S²× S³ (Shu-Cheng Chang, Chin-Tung Wu and Liuyang Zhang, 2023)6Chang 2023 Differential topology of the spaces of asymptotically stable vector fields and Lyapunov functions (Matthew D. Kvalheim, 2025)8Kvalheim 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Notes on Perelman's papers

    Bruce Kleiner and John LottThe canon PDF

    Topics Surgery, 3-manifolds · Same author Bruce Kleiner · Shared terms “Perelman”

  2. Singular Ricci flows I

    Bruce Kleiner and John LottThe canon PDF

    Topics Surgery, 3-manifolds · Same author Bruce Kleiner · Shared terms “compact Riemannian”

  3. The Smale conjecture and min-max theory

    Daniel Ketover and Yevgeny Liokumovich PDF

    Topics Smale conjecture & diffeomorphisms

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

    Richard H Bamler and Bruce KleinerWider flows PDF

    Topics Singularities · Same authors Richard H. Bamler, Bruce Kleiner

  5. Structure theory of non-collapsed limits of Ricci flows

    Richard H BamlerThe canon PDF

    Topics 3-manifolds, Singularities · Same author Richard H. Bamler · Shared terms “Perelman”

  6. The Smale Conjecture and Minimal Legendrian Graph in

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

    Topics Smale conjecture & diffeomorphisms, Singularities, Stability & uniqueness

  7. Cobordism, Singularities and the Ricci Flow Conjecture

    David Martín Velázquez, Davide De Biasio and Dieter Lust PDF

    Topics Surgery, Singularities

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

    Matthew D. Kvalheim PDF

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

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: uniqueness and stability of Ricci flow through singularities.

Foundational paper in the Weak & singular flows guide: Bamler–Kleiner: uniqueness and stability of singular flows.

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 usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Surgery Stop the flow just before a neck pinches, cut along the thin neck, glue standard caps onto the two ends, and restart. Start with: Four-manifolds with positive isotropic curvature (Hamilton, 1997) 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)

In the guidebook

  1. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

    Every high-curvature point looks like a neck or a cap.

  2. 12B.7
    The Singular Time: Horns and Strong NecksBook 12B · κ-Solutions and Surgery

    The limit metric at a singular time, the set Ω where curvature stays bounded, ε-horns and their ends, and strong δ-necks deep in a horn, following Perelman II §3 and Morgan–Tian chapter 11.

  3. 12B.12
    Canonical Neighbourhoods After SurgeryBook 12B · κ-Solutions and Surgery

    The strong canonical neighbourhood assumption justified across surgeries: the blow-up argument, the three cases for a limit that meets a surgery, and the choice of δ (Perelman II §5; Morgan–Tian chapter 17).