Paper · May 22, 2026

Geometrisation of 3-manifolds

Bruno Martelli

math.GTarXiv:2605.23679
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Abstract

The geometrisation theorem of 3-manifolds was conjectured by Thurston the 1980s and proved by Perelman in the 2000s. This is an overview on the subject. We explain the content of the theorem and describe its effects in various situations.

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Submitted
May 22, 2026
Subjects
math.GT, math.DG

Comments: 27 pages, 26 figures

Geometrization3-manifolds

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)5Perelman 2002 Geometrization in Geometry (Izabella Muraro de Freitas and Álvaro Krüger Ramos, 2022)1de Freitas 2022 Large-scale geometry of the Universe (Yassir Awwad and Tomislav Prokopec, 2022)2Awwad 2022 Modified mean curvature flow and CMC foliation conjecture in almost Fuchsian manifolds (Zheng Huang, Longzhi Lin and Zhou Zhang, 2023)6Huang 2023 Combinatorial Ricci Flow and Thurston's Triangulation Conjecture (Feng Ke and Ge Huabin, 2025)4Ke 2025 Sigma model renormalisation group flows, singularities and some remarks on cosmology (Georgios Papadopoulos, 2025)8Papadopoulos 2025 Modifications of CMB Temperature and Polarization Quadrupole Signals in Thurston Spacetimes (Tanay Gupta, Sukanta Panda and Rajib Saha, 2026)7Gupta 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. Geometrization in Geometry

    Izabella Muraro de Freitas and Álvaro Krüger Ramos

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

  2. Large-scale geometry of the Universe

    Yassir Awwad and Tomislav Prokopec

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

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

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

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

  4. Combinatorial Ricci Flow and Thurston's Triangulation Conjecture

    Feng Ke and Ge Huabin

    Topics Geometrization, 3-manifolds · Shared terms “Thurston”

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

    Grisha PerelmanThe canon

    Topics Geometrization, 3-manifolds · Shared terms “Thurston”

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

    Zheng Huang, Longzhi Lin and Zhou Zhang

    Topics Geometrization · Shared terms “Thurston”

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

    Tanay Gupta, Sukanta Panda and Rajib Saha

    Topics Geometrization · Shared terms “Thurston”

  8. Sigma model renormalisation group flows, singularities and some remarks on cosmology

    Georgios Papadopoulos

    Topics Geometrization · Shared terms “geometrisation”

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) 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.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.