The canon · Paper · May 25, 2006

Notes on Perelman's papers

Bruce Kleiner, John Lott

math.DGarXiv:math/0605667v5Geom. Topol. 12 (2008) 2587-2855
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Abstract

Kleiner–Lott. Line-by-line notes filling in the details of Perelman's papers. The standard companion.

These are detailed notes on Perelman's papers "The entropy formula for the Ricci flow and its geometric applications" and "Ricci flow with surgery on three-manifolds".

Comments: 216 pages, minor corrections made

Ricci flowSurgery3-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)3Perelman 2003 Finite extinction time for the solutions to the Ricci flow on certain three-manifolds (Grisha Perelman, 2003)8Perelman 2003 Singular Ricci flows I (Bruce Kleiner and John Lott, 2014)1Kleiner 2014 Uniqueness and stability of Ricci flow through singularities (Richard H. Bamler and Bruce Kleiner, 2017)2Bamler 2017 A note on blowup limits in 3d Ricci flow (Beomjun Choi and Robert Haslhofer, 2021)7Choi 2021 A note on Alexandrov immersed mean curvature flow (Ben Lambert and Elena Mäder-Baumdicker, 2022)4Lambert 2022 Towards a Taub-Bolt to Taub-NUT via Ricci flow with surgery (John Hughes, 2025)5Hughes 2025 A Lean Formalization of the Hamilton–Perelman Proof of the Three-Dimensional Poincaré Conjecture (Ziyang Qin, Yuan Liao, Ayush Khaitan et al., 2026)6Qin 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Singular Ricci flows I

    Bruce Kleiner and John LottThe canon PDF

    Topics Surgery, 3-manifolds · Same authors Bruce Kleiner, John Lott

  2. Uniqueness and stability of Ricci flow through singularities

    Richard H. Bamler and Bruce KleinerThe canon PDF

    Topics Surgery, 3-manifolds · Same author Bruce Kleiner · Shared terms “Perelman”

  3. Ricci flow with surgery on three-manifolds

    Grisha PerelmanThe canon PDF

    Topics Surgery, 3-manifolds

  4. A note on Alexandrov immersed mean curvature flow

    Ben Lambert and Elena Mäder-BaumdickerWider flows PDF

    Topics Surgery

  5. Towards a Taub-Bolt to Taub-NUT via Ricci flow with surgery

    John Hughes PDF

    Topics Surgery

  6. A Lean Formalization of the Hamilton–Perelman Proof of the Three-Dimensional Poincaré Conjecture

    Ziyang Qin, Yuan Liao, Ayush Khaitan et al. PDF

    Topics Surgery, 3-manifolds · Shared terms “Perelman”

  7. A note on blowup limits in 3d Ricci flow

    Beomjun Choi and Robert Haslhofer PDF

    Topics 3-manifolds · Shared terms “Perelman”

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

    Grisha PerelmanThe canon PDF

    Topics Surgery, 3-manifolds

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 Ancient solutions guide: Kleiner–Lott's notes, with detailed proofs of Perelman's results on κ-solutions.

Survey in the Perelman entropy guide: Kleiner–Lott: detailed proofs of Perelman's monotonicity formulas.

Survey in the (Non)collapsing guide: Kleiner–Lott: Perelman's non-collapsing proofs in full detail.

Survey in the Poincaré conjecture guide: Kleiner–Lott: line-by-line notes on Perelman's papers.

Survey in the Reduced volume guide: Kleiner–Lott: complete proofs of the properties of and reduced volume.

Survey in the Surgery guide: Kleiner–Lott: detailed notes on Perelman's papers, including every step of surgery.

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) Surgery Stop the flow just before a neck pinches, cut along the thin neck, glue standard caps onto the two ends, and restart. Start with: Four-manifolds with positive isotropic curvature (Hamilton, 1997) 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. 11B.7
    Hamilton’s Program in 2002Book 11B · Solitons, Compactness and Singularities

    What was proved, what was missing, and where Perelman came in.

  2. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

    Every high-curvature point looks like a neck or a cap.

  3. 12B.7
    The Singular Time: Horns and Strong NecksBook 12B · κ-Solutions and Surgery

    The limit metric at a singular time, the set Ω where curvature stays bounded, ε-horns and their ends, and strong δ-necks deep in a horn, following Perelman II §3 and Morgan–Tian chapter 11.