The canon · Paper · Jul 25, 2006

Ricci Flow and the Poincare Conjecture

John W. Morgan, Gang Tian

math.DGarXiv:math/0607607v2
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Abstract

Morgan–Tian. A complete, book-length proof of the Poincaré conjecture (Clay Mathematics Monographs).

This manuscript contains a detailed proof of the Poincare Conjecture. The arguments we present here are expanded versions of the ones given by Perelman in his three preprints posted in 2002 and 2003. This is a revised version taking in account the comments of the referees and others. It has been reformatted in the AMS book style.

Comments: 493 pages with over 30 figures and 3 pages of front material

Ricci flowPoincaré conjecture

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 Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture (Huai-Dong Cao and Xi-Ping Zhu, 2006)3Cao 2006 Completion of the Proof of the Geometrization Conjecture (John Morgan and Gang Tian, 2008)1Morgan 2008 Product structure and regularity theorem for totally nonnegative flag varieties (Huanchen Bao and Xuhua He, 2022)8Bao 2022 On 4-dimensional smooth Poincare conjecture (Selman Akbulut, 2022)2Akbulut 2022 Rigor with Machine Learning from Field Theory to the Poincaré Conjecture (Sergei Gukov, James Halverson and Fabian Ruehle, 2024)6Gukov 2024 Moving Manifolds and the Poincare Conjecture (David V. Svintradze, 2025)5Svintradze 2025 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)4Qin 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 · Same authors John W. Morgan, Gang Tian · Shared terms “preprints”, “Perelman”

  2. On 4-dimensional smooth Poincare conjecture

    Selman Akbulut PDF

    Topics Poincaré conjecture

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

    Huai-Dong Cao and Xi-Ping ZhuThe canon PDF

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

  5. Moving Manifolds and the Poincare Conjecture

    David V. Svintradze PDF

    Topics Poincaré conjecture

  6. Rigor with Machine Learning from Field Theory to the Poincaré Conjecture

    Sergei Gukov, James Halverson and Fabian Ruehle PDF

    Topics Poincaré conjecture · Shared terms “Perelman”

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

    Olga Kharlampovich and Alina Vdovina PDF

    Topics Poincaré conjecture · Shared terms “Perelman”

  8. Product structure and regularity theorem for totally nonnegative flag varieties

    Huanchen Bao and Xuhua He PDF

    Topics Poincaré conjecture

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 Poincaré conjecture guide: Morgan–Tian: *Ricci Flow and the Poincaré Conjecture*, a complete book-length proof.

Survey in the Reduced volume guide: Morgan–Tian: a textbook treatment of reduced length and its applications to κ-solutions.

Survey in the Surgery guide: Morgan–Tian: a book-length, complete account of Ricci flow with surgery.

Survey in the 3-manifolds guide: Morgan–Tian: *Ricci Flow and 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) 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)

In the guidebook

  1. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  2. 7A.12
    The Poincaré Conjecture, PreciselyBook 7A · Topology and the Fundamental Group

    Every word of the statement, and why each hypothesis is needed.

  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.