The canon · Paper · Dec 3, 2006

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

Huai-Dong Cao, Xi-Ping Zhu

math.DGarXiv:math/0612069
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Abstract

Cao–Zhu. An independent exposition of the Hamilton–Perelman proof of geometrization.

In this paper, we provide an essentially self-contained and detailed account of the fundamental works of Hamilton and the recent breakthrough of Perelman on the Ricci flow and their application to the geometrization of three-manifolds. In particular, we give a detailed exposition of a complete proof of the Poincaré conjecture due to Hamilton and Perelman.

Comments: This is a revised version of the article by the same authors that originally appeared in Asian J. Math., {\bf 10}({\bf 2}) (2006), 165–492

Ricci flowPoincaré conjectureGeometrization3-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)8Perelman 2002 Ricci Flow and the Poincare Conjecture (John W. Morgan and Gang Tian, 2006)3Morgan 2006 Completion of the Proof of the Geometrization Conjecture (John Morgan and Gang Tian, 2008)1Morgan 2008 On 4-dimensional smooth Poincare conjecture (Selman Akbulut, 2022)5Akbulut 2022 Geometrization in Geometry (Izabella Muraro de Freitas and Álvaro Krüger Ramos, 2022)6de Freitas 2022 Equations in Products of Free Groups and 3-Manifold Groups, I (Olga Kharlampovich and Alina Vdovina, 2026)7Kharlampovic 2026 A Lean Formalization of the Hamilton–Perelman Proof of the Three-Dimensional Poincaré Conjecture (Ziyang Qin, Yuan Liao, Ayush Khaitan et al., 2026)2Qin 2026 A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture (Yuan Liao, Ziyang Qin, Jui-Hui Chung et al., 2026)4Liao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Completion of the Proof of the Geometrization Conjecture

    John Morgan and Gang TianThe canon PDF

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

  2. 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, 3-manifolds · Shared terms “Perelman proof”, “Hamilton”

  3. Ricci Flow and the Poincare Conjecture

    John W. Morgan and Gang TianThe canon PDF

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

  4. 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 proof”, “Hamilton”

  5. On 4-dimensional smooth Poincare conjecture

    Selman Akbulut PDF

    Topics Poincaré conjecture

  6. Geometrization in Geometry

    Izabella Muraro de Freitas and Álvaro Krüger Ramos PDF

    Topics Geometrization · Shared terms “Poincaré conjecture”, “breakthrough”, “Hamilton”

  7. Equations in Products of Free Groups and 3-Manifold Groups, I

    Olga Kharlampovich and Alina Vdovina PDF

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

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

    Grisha PerelmanThe canon PDF

    Topics Geometrization, 3-manifolds · Shared terms “geometrization conjecture”, “Hamilton”

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.

Survey in the Geometrization guide: Cao–Zhu: an exposition of the Hamilton–Perelman proof of both conjectures.

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

    The fate of each piece under the flow.

  2. 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.

  3. 10A.6
    Necks, Caps and Their TopologyBook 10A · Three-Manifolds and Geometrization

    What an ε-neck is, how necks overlap and chain into tubes, and which manifolds necks and caps can build.