The canon · Paper · Aug 10, 2014

Singular Ricci flows I

Bruce Kleiner, John Lott

math.DGarXiv:1408.2271v3
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Abstract

Kleiner–Lott. A canonical Ricci flow through singularities — no surgery parameters needed.

We introduce singular Ricci flows, which are Ricci flow spacetimes subject to certain asymptotic conditions. We consider the behavior of Ricci flow with surgery starting from a fixed initial compact Riemannian 3-manifold, as the surgery parameter varies. We prove that the flow with surgery subconverges to a singular Ricci flow as the surgery parameter tends to zero. We establish a number of geometric and analytical properties of singular Ricci flows.

Comments: Final version. 76 pages. Sections 8 and 9 of the previous version are moved to a separate paper

Ricci flowSurgery3-manifoldsWeak & singular flows

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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 Ricci flow with surgery on three-manifolds (Grisha Perelman, 2003)5Perelman 2003 Notes on Perelman's papers (Bruce Kleiner and John Lott, 2006)1Kleiner 2006 Uniqueness and stability of Ricci flow through singularities (Richard H. Bamler and Bruce Kleiner, 2017)2Bamler 2017 Uniqueness and stability of singular Ricci flows in higher dimensions (Robert Haslhofer, 2021)6Haslhofer 2021 Canonical surgeries in rotationally invariant Ricci flow (Timothy Buttsworth, Maximilien Hallgren and Yongjia Zhang, 2022)3Buttsworth 2022 Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flows (Albert Chau and Adam Martens, 2023)4Chau 2023 Singular Ricci Flows on surfaces with boundary and positive scalar curvature (Jean C. Cortissoz and Juan J. Villamarín, 2023)7Cortissoz 2023 Towards a Taub-Bolt to Taub-NUT via Ricci flow with surgery (John Hughes, 2025)8Hughes 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 authors Bruce Kleiner, John Lott

  2. Uniqueness and stability of Ricci flow through singularities

    Richard H. Bamler and Bruce KleinerThe canon PDF

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

  3. Canonical surgeries in rotationally invariant Ricci flow

    Timothy Buttsworth, Maximilien Hallgren and Yongjia Zhang PDF

    Topics Surgery · Shared terms “surgery starting”, “flow spacetime”

  4. Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flows

    Albert Chau and Adam Martens PDF

    Topics 3-manifolds, Weak & singular flows

  5. Ricci flow with surgery on three-manifolds

    Grisha PerelmanThe canon PDF

    Topics Surgery, 3-manifolds

  6. Uniqueness and stability of singular Ricci flows in higher dimensions

    Robert Haslhofer PDF

    Topics 3-manifolds, Weak & singular flows

  7. Singular Ricci Flows on surfaces with boundary and positive scalar curvature

    Jean C. Cortissoz and Juan J. Villamarín PDF

    Topics Weak & singular flows

  8. Towards a Taub-Bolt to Taub-NUT via Ricci flow with surgery

    John Hughes PDF

    Topics Surgery

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: Kleiner–Lott: singular Ricci flows.

Foundational paper in the Surgery guide: Kleiner–Lott: singular Ricci flows, the limit of surgery as the scale goes to zero.

Foundational paper in the Weak & singular flows guide: Kleiner–Lott: singular Ricci 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) 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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 11B.7
    Hamilton’s Program in 2002Book 11B · Solitons, Compactness and Singularities

    What was proved, what was missing, and where Perelman came in.

  2. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

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

  3. 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.