The canon · Paper · Feb 27, 2020

Uniqueness of compact ancient solutions to three-dimensional Ricci flow

Simon Brendle, Panagiota Daskalopoulos, Natasa Sesum

math.DGarXiv:2002.12240v2
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Abstract

Brendle–Daskalopoulos–Sesum. The compact case on S³: shrinking round spheres or Perelman's Type II solution.

In this paper, we study the classification of -noncollapsed ancient solutions to three-dimensional Ricci flow on . We prove that such a solution is either isometric to a family of shrinking round spheres, or the Type II ancient solution constructed by Perelman.

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Submitted
Feb 27, 2020
Last revised
Apr 30, 2021 (v2)
Subjects
math.DG, math.AP
DOI
10.1007/s00222-021-01054-0

Comments: This paper supersedes the paper arXiv:1904.07835 by Brendle and the paper arXiv:1907.01928 by Daskalopoulos and Sesum. We also remove the assumption on reflection symmetry that was present in the latter paper

Ricci flowAncient solutionsSingularities(Non)collapsing3-manifoldsStability & uniqueness

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 Ancient solutions to the Ricci flow in dimension 3 (S. Brendle, 2018)2Brendle 2018 Singularity models in the three-dimensional Ricci flow (S. Brendle, 2022)1Brendle 2022 Classification of bubble-sheet ovals in ℝ⁴ (Beomjun Choi, Panagiota Daskalopoulos, Wenkui Du et al., 2022)6Choi 2022 Dynamics of Convex Mean Curvature Flow (Sigurd Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2023)4Angenent 2023 Mean curvature flow near a peanut solution (Sigurd Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2025)5Angenent 2025 Unique asymptotics of SO(k)× SO(n-k+1) symmetric ancient ovals of Ricci flow (Panagiota Daskalopoulos, Wenkui Du, Natasa Sesum et al., 2026)3Daskalopoulo 2026 Formation and structural stability of nondegenerate neckpinches (Kyeongsu Choi, Panagiota Daskalopoulos and Natasa Sesum, 2026)7Choi 2026 Ricci flow on manifolds with positive isotropic curvature in all dimensions (Simon Brendle and Raphael Tsiamis, 2026)8Brendle 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Singularity models in the three-dimensional Ricci flow

    S. Brendle

    Topics (Non)collapsing, Ancient solutions, 3-manifolds · Same author Simon Brendle · Shared terms “dimensional Ricci”, “Perelman”, “noncollapsed”

  2. Ancient solutions to the Ricci flow in dimension 3

    S. BrendleThe canon

    Topics Ancient solutions, 3-manifolds, Singularities · Same author Simon Brendle · Shared terms “Perelman”

  3. Unique asymptotics of symmetric ancient ovals of Ricci flow

    Panagiota Daskalopoulos, Wenkui Du, Natasa Sesum et al.

    Topics Ancient solutions, Singularities, Stability & uniqueness · Same authors Panagiota Daskalopoulos, Natasa Sesum · Shared terms “shrinking round”, “round sphere”, “Perelman”

  4. Dynamics of Convex Mean Curvature Flow

    Sigurd Angenent, Panagiota Daskalopoulos and Natasa SesumWider flows

    Topics Ancient solutions, Stability & uniqueness · Same authors Panagiota Daskalopoulos, Natasa Sesum

  5. Mean curvature flow near a peanut solution

    Sigurd Angenent, Panagiota Daskalopoulos and Natasa SesumWider flows

    Topics Ancient solutions, Singularities, Stability & uniqueness · Same authors Panagiota Daskalopoulos, Natasa Sesum

  6. Classification of bubble-sheet ovals in

    Beomjun Choi, Panagiota Daskalopoulos, Wenkui Du et al.Wider flows

    Topics Ancient solutions · Same authors Panagiota Daskalopoulos, Natasa Sesum

  7. Formation and structural stability of nondegenerate neckpinches

    Kyeongsu Choi, Panagiota Daskalopoulos and Natasa SesumWider flows

    Topics 3-manifolds, Singularities, Stability & uniqueness · Same authors Panagiota Daskalopoulos, Natasa Sesum

  8. Ricci flow on manifolds with positive isotropic curvature in all dimensions

    Simon Brendle and Raphael Tsiamis

    Topics (Non)collapsing, Ancient solutions · Same author Simon Brendle · Shared terms “noncollapsed ancient”

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: Brendle–Daskalopoulos–Šešum: uniqueness of compact ancient solutions, completing the 3D classification.

Foundational paper in the Ricci flow guide: Brendle–Daskalopoulos–Šešum: compact κ-noncollapsed ancient solutions on .

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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, 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)

In the guidebook

  1. 12B.4
    Compactness and Universal Estimates for κ-SolutionsBook 12B · κ-Solutions and Surgery

    Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.

  2. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

  3. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.