The canon · Paper · Sep 23, 2008

Completion of the Proof of the Geometrization Conjecture

John Morgan, Gang Tian

math.DGarXiv:0809.4040
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Abstract

Morgan–Tian. Completing the proof of Thurston's geometrization conjecture.

This article is a sequel to the book `Ricci Flow and the Poincare Conjecture' by the same authors. Using the main results of that book we establish the Geometrization Conjecture for all compact, orientable three-manifolds following the approach indicated by Perelman in his preprints on the subject. This approach is to study the collapsed part of the manifold as time goes to infinity in a Ricci flow with surgery. The main technique for this study is the theory of Alexandrov spaces. This theory gives local models for the collapsed part of the manifold. These local models can be glued together to prove that the collapsed part of the manifold is a graph manifold with incompressible boundary. From this and previous results, geometrization follows easily.

Comments: 84 pages

Ricci flowSurgeryPoincaré conjectureGeometrization3-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 Ricci flow with surgery on three-manifolds (Grisha Perelman, 2003)8Perelman 2003 Ricci Flow and the Poincare Conjecture (John W. Morgan and Gang Tian, 2006)1Morgan 2006 Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture (Huai-Dong Cao and Xi-Ping Zhu, 2006)2Cao 2006 On 4-dimensional smooth Poincare conjecture (Selman Akbulut, 2022)6Akbulut 2022 Geometrization in Geometry (Izabella Muraro de Freitas and Álvaro Krüger Ramos, 2022)5de Freitas 2022 Moving Manifolds and the Poincare Conjecture (David V. Svintradze, 2025)7Svintradze 2025 A Lean Formalization of the Hamilton–Perelman Proof of the Three-Dimensional Poincaré Conjecture (Ziyang Qin, Yuan Liao, Ayush Khaitan et al., 2026)4Qin 2026 A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture (Yuan Liao, Ziyang Qin, Jui-Hui Chung et al., 2026)3Liao 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 and the Poincare Conjecture

    John W. Morgan and Gang TianThe canon PDF

    Topics Poincaré conjecture · Same authors John Morgan, Gang Tian · Shared terms “preprints”, “Perelman”

  2. Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture

    Huai-Dong Cao and Xi-Ping ZhuThe canon PDF

    Topics Poincaré conjecture, Geometrization, 3-manifolds · Shared terms “Perelman”

  3. A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture

    Yuan Liao, Ziyang Qin, Jui-Hui Chung et al. PDF

    Topics Geometrization · Shared terms “geometrization conjecture”, “Perelman”

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

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

    Topics Poincaré conjecture, Surgery, 3-manifolds · Shared terms “Perelman”, “together”

  5. Geometrization in Geometry

    Izabella Muraro de Freitas and Álvaro Krüger Ramos PDF

    Topics Geometrization · Shared terms “Poincaré conjecture”, “Perelman”

  6. On 4-dimensional smooth Poincare conjecture

    Selman Akbulut PDF

    Topics Poincaré conjecture

  7. Moving Manifolds and the Poincare Conjecture

    David V. Svintradze PDF

    Topics Poincaré conjecture, Surgery

  8. Ricci flow with surgery on three-manifolds

    Grisha PerelmanThe canon PDF

    Topics Surgery, Discrete & graph Ricci, 3-manifolds

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: Morgan–Tian: completing the proof of the geometrization 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) 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) Poincaré conjecture Every closed, simply connected 3-manifold is homeomorphic to the 3-sphere. Poincaré posed the question in 1904, and Perelman proved it in 2002–03… Start with: Generalized Poincare's Conjecture in Dimensions Greater Than Four (Smale, 1961) Geometrization Thurston's conjecture: every closed 3-manifold can be cut along spheres and tori into pieces that each carry one of eight model geometries. Start with: Three dimensional manifolds, Kleinian groups and hyperbolic geometry (Thurston, 1982)

In the guidebook

  1. 10A.10
    Geometrization and Ricci FlowBook 10A · Three-Manifolds and Geometrization

    The fate of each piece under the flow.

  2. 11B.7
    Hamilton’s Program in 2002Book 11B · Solitons, Compactness and Singularities

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

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.