Paper · Oct 18, 2022

Geometrization in Geometry

Izabella Muraro de Freitas, Álvaro Krüger Ramos

math.DGarXiv:2210.09781
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Abstract

So far, the most magnificent breakthrough in mathematics in the 21st century is the Geometrization Theorem, a bold conjecture by William Thurston (generalizing Poincaré's Conjecture) and proved by Grigory Perelman, based on the program suggested by Richard Hamilton. In this survey article, we will explain the statement of this result, also presenting some examples of how it can be used to obtain interesting results in differential geometry.

Comments: To appear in Matemática Contemporânea, special issue in honor of Professor Renato Tribuzy on the occasion of his 75th birthday

Geometrization

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 The entropy formula for the Ricci flow and its geometric applications (Grisha Perelman, 2002)4Perelman 2002 Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture (Huai-Dong Cao and Xi-Ping Zhu, 2006)2Cao 2006 Completion of the Proof of the Geometrization Conjecture (John Morgan and Gang Tian, 2008)7Morgan 2008 Large-scale geometry of the Universe (Yassir Awwad and Tomislav Prokopec, 2022)6Awwad 2022 Modified mean curvature flow and CMC foliation conjecture in almost Fuchsian manifolds (Zheng Huang, Longzhi Lin and Zhou Zhang, 2023)8Huang 2023 Modifications of CMB Temperature and Polarization Quadrupole Signals in Thurston Spacetimes (Tanay Gupta, Sukanta Panda and Rajib Saha, 2026)3Gupta 2026 Geometrisation of 3-manifolds (Bruno Martelli, 2026)5Martelli 2026 A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture (Yuan Liao, Ziyang Qin, Jui-Hui Chung et al., 2026)1Liao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 “Thurston”, “Hamilton”, “Perelman”

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

    Huai-Dong Cao and Xi-Ping ZhuThe canon PDF

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

  3. Modifications of CMB Temperature and Polarization Quadrupole Signals in Thurston Spacetimes

    Tanay Gupta, Sukanta Panda and Rajib Saha PDF

    Topics Geometrization · Shared terms “William Thurston”

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

    Grisha PerelmanThe canon PDF

    Topics Geometrization · Shared terms “Richard Hamilton”, “Thurston”

  5. Geometrisation of 3-manifolds

    Bruno Martelli PDF

    Topics Geometrization · Shared terms “Thurston”, “Perelman”

  6. Large-scale geometry of the Universe

    Yassir Awwad and Tomislav Prokopec PDF

    Topics Geometrization · Shared terms “Thurston”, “Perelman”

  7. Completion of the Proof of the Geometrization Conjecture

    John Morgan and Gang TianThe canon PDF

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

  8. Modified mean curvature flow and CMC foliation conjecture in almost Fuchsian manifolds

    Zheng Huang, Longzhi Lin and Zhou Zhang PDF

    Topics Geometrization · Shared terms “Thurston”

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.

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.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  2. 10A.8
    Thurston’s Eight GeometriesBook 10A · Three-Manifolds and Geometrization

    The model geometries, the geometrization conjecture, and why it implies Poincaré.

  3. 10A.9
    Hyperbolic Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    Ideal tetrahedra, Mostow rigidity, and the thick–thin decomposition.