The canon · Paper · Mar 10, 2003

Ricci flow with surgery on three-manifolds

Grisha Perelman

math.DGarXiv:math/0303109
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As on arXiv

Abstract

Perelman II. Ricci flow with surgery on 3-manifolds — the engine of the geometrization proof.

This is a technical paper, which is a continuation of math.DG/0211159. Here we construct Ricci flow with surgeries and verify most of the assertions, made in section 13 of that e-print; the exceptions are (1) the statement that manifolds that can collapse with local lower bound on sectional curvature are graph manifolds - this is deferred to a separate paper, since the proof has nothing to do with the Ricci flow, and (2) the claim on the lower bound for the volume of maximal horns and the smoothness of solutions from some time on, which turned out to be unjustified and, on the other hand, irrelevant for the other conclusions.

Comments: 22 pages

Ricci flowSurgery3-manifoldsDiscrete & graph Ricci

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 Finite extinction time for the solutions to the Ricci flow on certain three-manifolds (Grisha Perelman, 2003)1Perelman 2003 Notes on Perelman's papers (Bruce Kleiner and John Lott, 2006)2Kleiner 2006 Completion of the Proof of the Geometrization Conjecture (John Morgan and Gang Tian, 2008)6Morgan 2008 Singular Ricci flows I (Bruce Kleiner and John Lott, 2014)3Kleiner 2014 The existence of inversive distance circle packing on polyhedral surface (Xiang Zhu, 2023)7Zhu 2023 A modified Ricci flow on arbitrary weighted graph (Jicheng Ma and Yunyan Yang, 2024)4Ma 2024 Evolution of weights on a connected finite graph (Jicheng Ma and Yunyan Yang, 2024)5Ma 2024 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. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds

    Grisha PerelmanThe canon PDF

    Topics Surgery, 3-manifolds · Same author Grisha Perelman · Shared terms “Math”

  2. Notes on Perelman's papers

    Bruce Kleiner and John LottThe canon PDF

    Topics Surgery, 3-manifolds

  3. Singular Ricci flows I

    Bruce Kleiner and John LottThe canon PDF

    Topics Surgery, 3-manifolds

  4. A modified Ricci flow on arbitrary weighted graph

    Jicheng Ma and Yunyan Yang PDF

    Topics Surgery, Discrete & graph Ricci

  5. Evolution of weights on a connected finite graph

    Jicheng Ma and Yunyan Yang PDF

    Topics Surgery, Discrete & graph Ricci

  6. Completion of the Proof of the Geometrization Conjecture

    John Morgan and Gang TianThe canon PDF

    Topics Surgery, Discrete & graph Ricci, 3-manifolds

  7. The existence of inversive distance circle packing on polyhedral surface

    Xiang Zhu PDF

    Topics Surgery, Discrete & graph Ricci

  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 Geometrization guide: Perelman II: long-time behaviour of Ricci flow with surgery.

Foundational paper in the Poincaré conjecture guide: Perelman II: Ricci flow with surgery on three-manifolds.

Foundational paper in the Surgery guide: Perelman: Ricci flow with surgery on three-manifolds.

Foundational paper in the 3-manifolds guide: Perelman: Ricci flow with surgery on three-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) 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) Discrete & graph Ricci Curvature and Ricci flow for graphs, networks and triangulated surfaces. It runs from Chow–Luo's combinatorial flow on circle packings to Ollivier's… Start with: Combinatorial Ricci Flows on Surfaces (Chow et al., 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.