The canon · Paper · Nov 6, 2018

Ancient solutions to the Ricci flow in dimension 3

S. Brendle

math.DGarXiv:1811.02559v3
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Abstract

Brendle. Classification of ancient κ-solutions in dimension three.

It is known from work of Perelman that any finite-time singularity of the Ricci flow on a compact three-manifold is modeled on an ancient -solution. We prove that the every noncompact ancient -solution in dimension is isometric to either the shrinking cylinders (or a quotient thereof), or the Bryant soliton.

Comments: revised version, to appear in Acta Mathematica

Ricci flowAncient solutionsSingularities3-manifoldsBoundaries & non-compact

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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 On Ricci flows with closed and smooth tangent flows (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)8Chan 2021 Singularity models in the three-dimensional Ricci flow (S. Brendle, 2022)1Brendle 2022 O(2)-symmetry of 3D steady gradient Ricci solitons (Yi Lai, 2022)7Lai 2022 Ancient solutions of Ricci flow with Type I curvature growth (Stephen Lynch and Andoni Royo Abrego, 2022)5Lynch 2022 Curvature estimates of ancient solutions to the mean curvature flow of higher codimension with convex Gauss image (Hongbing Qiu and Y. L. Xin, 2023)6Qiu 2023 A rigidity theorem of ancient solutions to the mean curvature flow in codimension one (Qun Chen and Hongbing Qiu, 2024)3Chen 2024 κ-solutions with the round cylinder as an asymptotic shrinker (Aprameya Girish Hebbar, 2026)2Hebbar 2026 Ricci flow on manifolds with positive isotropic curvature in all dimensions (Simon Brendle and Raphael Tsiamis, 2026)4Brendle 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 PDF

    Topics Ancient solutions, 3-manifolds, Singularities · Same author S. Brendle · Shared terms “time singularity”, “Perelman”, “finite time”

  2. -solutions with the round cylinder as an asymptotic shrinker

    Aprameya Girish Hebbar PDF

    Topics Ancient solutions, Boundaries & non-compact · Shared terms “shrinking cylinder”, “Bryant”, “Perelman”

  3. A rigidity theorem of ancient solutions to the mean curvature flow in codimension one

    Qun Chen and Hongbing QiuWider flows PDF

    Topics Ancient solutions, Boundaries & non-compact · Shared terms “noncompact ancient”

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

    Simon Brendle and Raphael Tsiamis PDF

    Topics Ancient solutions, Boundaries & non-compact · Same author S. Brendle

  5. Ancient solutions of Ricci flow with Type I curvature growth

    Stephen Lynch and Andoni Royo Abrego PDF

    Topics Ancient solutions, Boundaries & non-compact, Singularities

  6. Curvature estimates of ancient solutions to the mean curvature flow of higher codimension with convex Gauss image

    Hongbing Qiu and Y. L. XinWider flows PDF

    Topics Ancient solutions, Boundaries & non-compact · Shared terms “noncompact ancient”

  7. O(2)-symmetry of 3D steady gradient Ricci solitons

    Yi Lai PDF

    Topics Ancient solutions, 3-manifolds · Shared terms “Bryant soliton”

  8. On Ricci flows with closed and smooth tangent flows

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang PDF

    Topics Ancient solutions, Singularities · Shared terms “time singularity”, “finite time”

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: non-compact ancient κ-solutions in dimension three.

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

    Necks and caps, compactness, and universal estimates.

  2. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  3. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.