The canon · Paper · Jul 17, 2003

Finite extinction time for the solutions to the Ricci flow on certain three-manifolds

Grisha Perelman

math.DGarXiv:math/0307245
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Abstract

Perelman III. A shortcut to the Poincaré conjecture: the flow on simply connected manifolds goes extinct in finite time.

Let M be a closed oriented three-manifold, whose prime decomposition contains no aspherical factors. We show that for any initial riemannian metric on M the solution to the Ricci flow with surgery, defined in our previous paper math.DG/0303109, becomes extinct in finite time. The proof uses a version of the minimal disk argument from 1999 paper by Richard Hamilton, and a regularization of the curve shortening flow, worked out by Altschuler and Grayson.

Comments: 7 pages

Ricci flowCurve shorteningSingularitiesSurgery3-manifolds

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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 The entropy formula for the Ricci flow and its geometric applications (Grisha Perelman, 2002)3Perelman 2002 Ricci flow with surgery on three-manifolds (Grisha Perelman, 2003)1Perelman 2003 Notes on Perelman's papers (Bruce Kleiner and John Lott, 2006)4Kleiner 2006 Singular Ricci flows I (Bruce Kleiner and John Lott, 2014)5Kleiner 2014 Free boundary flow with surgery (Robert Haslhofer, 2023)8Haslhofer 2023 Finite extinction time of a family of homogeneous Ricci flows (Roberto Araujo, 2025)2Araujo 2025 Towards a Taub-Bolt to Taub-NUT via Ricci flow with surgery (John Hughes, 2025)7Hughes 2025 A Lean Formalization of the Hamilton–Perelman Proof of the Three-Dimensional Poincaré Conjecture (Ziyang Qin, Yuan Liao, Ayush Khaitan et al., 2026)6Qin 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci flow with surgery on three-manifolds

    Grisha PerelmanThe canon PDF

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

  2. Finite extinction time of a family of homogeneous Ricci flows

    Roberto Araujo PDF

    Topics Ricci flow · Shared terms “finite extinction”, “extinction time”

  3. The entropy formula for the Ricci flow and its geometric applications

    Grisha PerelmanThe canon PDF

    Topics 3-manifolds, Singularities · Same author Grisha Perelman · Shared terms “Richard Hamilton”, “finite time”

  4. Notes on Perelman's papers

    Bruce Kleiner and John LottThe canon PDF

    Topics Surgery, 3-manifolds

  5. Singular Ricci flows I

    Bruce Kleiner and John LottThe canon PDF

    Topics Surgery, 3-manifolds

  6. A Lean Formalization of the Hamilton–Perelman Proof of the Three-Dimensional Poincaré Conjecture

    Ziyang Qin, Yuan Liao, Ayush Khaitan et al. PDF

    Topics Surgery, 3-manifolds, Singularities · Shared terms “extinction”, “Hamilton”, “finite time”

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

    John Hughes PDF

    Topics Surgery, Singularities

  8. Free boundary flow with surgery

    Robert HaslhoferWider flows PDF

    Topics Surgery, Singularities · Shared terms “becomes extinct”, “minimal disk”, “finite time”

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 Minimal & CMC surfaces guide: Perelman: finite extinction time for the Ricci flow on certain 3-manifolds.

Foundational paper in the Poincaré conjecture guide: Perelman III: finite extinction time, the shortcut to the Poincaré conjecture.

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) Curve shortening Wider flows Plane curves moving by their curvature. Convex curves shrink to round points (Gage–Hamilton), and every embedded curve becomes convex first (Grayson). Start with: The heat equation shrinking convex plane curves (Gage et al., 1986) 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)

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).