The canon · Paper · Nov 11, 2002

The entropy formula for the Ricci flow and its geometric applications

Grisha Perelman

math.DGarXiv:math/0211159
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Abstract

Perelman I. The 𝓦-entropy, no local collapsing, and the canonical neighborhood picture. Start here after Hamilton.

We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric applications are given. In particular, (1) Ricci flow, considered on the space of riemannian metrics modulo diffeomorphism and scaling, has no nontrivial periodic orbits (that is, other than fixed points); (2) In a region, where singularity is forming in finite time, the injectivity radius is controlled by the curvature; (3) Ricci flow can not quickly turn an almost euclidean region into a very curved one, no matter what happens far away. We also verify several assertions related to Richard Hamilton's program for the proof of Thurston geometrization conjecture for closed three-manifolds, and give a sketch of an eclectic proof of this conjecture, making use of earlier results on collapsing with local lower curvature bound.

Comments: 39 pages

Ricci flowSingularities(Non)collapsingGeometrization3-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 Ricci flow with surgery on three-manifolds (Grisha Perelman, 2003)6Perelman 2003 Finite extinction time for the solutions to the Ricci flow on certain three-manifolds (Grisha Perelman, 2003)4Perelman 2003 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)8Morgan 2008 Geometrization in Geometry (Izabella Muraro de Freitas and Álvaro Krüger Ramos, 2022)2de Freitas 2022 Large-scale geometry of the Universe (Yassir Awwad and Tomislav Prokopec, 2022)5Awwad 2022 Geometrisation of 3-manifolds (Bruno Martelli, 2026)7Martelli 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 geometrization”, “geometrization conjecture”, “Hamilton”

  2. Geometrization in Geometry

    Izabella Muraro de Freitas and Álvaro Krüger Ramos PDF

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

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

    Huai-Dong Cao and Xi-Ping ZhuThe canon PDF

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

  4. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds

    Grisha PerelmanThe canon PDF

    Topics 3-manifolds, Singularities · Same author Grisha Perelman · Shared terms “Richard Hamilton”, “finite time”

  5. Large-scale geometry of the Universe

    Yassir Awwad and Tomislav Prokopec PDF

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

  6. Ricci flow with surgery on three-manifolds

    Grisha PerelmanThe canon PDF

    Topics 3-manifolds · Same author Grisha Perelman · Shared terms “local lower”

  7. Geometrisation of 3-manifolds

    Bruno Martelli PDF

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

  8. Completion of the Proof of the Geometrization Conjecture

    John Morgan and Gang TianThe canon PDF

    Topics Geometrization, 3-manifolds · Shared terms “geometrization 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.

Foundational paper in the Ancient solutions guide: Perelman: κ-solutions and the canonical neighbourhood theorem.

Foundational paper in the Eigenvalues & spectrum guide: Perelman: the -functional and its monotonicity.

Foundational paper in the Perelman entropy guide: Perelman: the entropy formula for the Ricci flow and its geometric applications.

Foundational paper in the Heat kernel guide: Perelman: the conjugate heat equation, its Harnack inequality and pseudolocality (sections 9–10).

Foundational paper in the (Non)collapsing guide: Perelman: the no local collapsing theorem (section 4).

Foundational paper in the Poincaré conjecture guide: Perelman I: the entropy formula and the canonical neighbourhood theorem.

Foundational paper in the Reduced volume guide: Perelman: the -length, reduced volume and its monotonicity (sections 6–7).

Foundational paper in the Ricci flow guide: Perelman's first preprint: the entropy formula and the ideas that finished Hamilton's program.

Foundational paper in the Ricci solitons guide: Perelman: shrinkers as singularity models, and the entropy that detects them.

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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) 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.