Paper · Sep 28, 2021

A note on blowup limits in 3d Ricci flow

Beomjun Choi, Robert Haslhofer

math.DGarXiv:2109.13875v2
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Abstract

We prove that Perelman's ancient ovals occur as blowup limit in 3d Ricci flow through singularities if and only if there is an accumulation of spherical singularities.

Ricci flowAncient solutionsSingularities3-manifoldsWeak & singular flows

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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 Hearing the shape of ancient noncollapsed flows in ℝ⁴ (Wenkui Du and Robert Haslhofer, 2021)7Du 2021 Ricci limit flows and weak solutions (Beomjun Choi and Robert Haslhofer, 2021)3Choi 2021 Uniqueness and stability of singular Ricci flows in higher dimensions (Robert Haslhofer, 2021)6Haslhofer 2021 Classification of bubble-sheet ovals in ℝ⁴ (Beomjun Choi, Panagiota Daskalopoulos, Wenkui Du et al., 2022)1Choi 2022 Enhanced profile estimates for ovals and translators (Kyeongsu Choi, Robert Haslhofer and Or Hershkovits, 2023)5Choi 2023 On κ-solutions and canonical neighborhoods in 4d Ricci flow (Robert Haslhofer, 2023)2Haslhofer 2023 Free boundary flow through cylindrical singularities (Yueheng Bao and Robert Haslhofer, 2026)8Bao 2026 Ancient cylindrical flows with fast or slow convergence (Kyeongsu Choi and Robert Haslhofer, 2026)4Choi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Classification of bubble-sheet ovals in

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

    Topics Ancient solutions · Same authors Beomjun Choi, Robert Haslhofer · Shared terms “ancient ovals”

  2. On -solutions and canonical neighborhoods in 4d Ricci flow

    Robert Haslhofer PDF

    Topics Ancient solutions, 3-manifolds, Weak & singular flows · Same author Robert Haslhofer · Shared terms “Perelman”

  3. Ricci limit flows and weak solutions

    Beomjun Choi and Robert Haslhofer PDF

    Topics Weak & singular flows, Singularities · Same authors Beomjun Choi, Robert Haslhofer

  4. Ancient cylindrical flows with fast or slow convergence

    Kyeongsu Choi and Robert HaslhoferWider flows PDF

    Topics Ancient solutions, Singularities · Same author Robert Haslhofer · Shared terms “ancient ovals”

  5. Enhanced profile estimates for ovals and translators

    Kyeongsu Choi, Robert Haslhofer and Or Hershkovits PDF

    Topics Ancient solutions · Same author Robert Haslhofer · Shared terms “ancient ovals”

  6. Uniqueness and stability of singular Ricci flows in higher dimensions

    Robert Haslhofer PDF

    Topics 3-manifolds, Weak & singular flows, Singularities · Same author Robert Haslhofer

  7. Hearing the shape of ancient noncollapsed flows in

    Wenkui Du and Robert HaslhoferWider flows PDF

    Topics Ancient solutions, 3-manifolds, Singularities · Same author Robert Haslhofer · Shared terms “ancient ovals”

  8. Free boundary flow through cylindrical singularities

    Yueheng Bao and Robert HaslhoferWider flows PDF

    Topics Ancient solutions, Weak & singular flows, Singularities · Same author Robert Haslhofer

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.

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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

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

    Necks and caps, compactness, and universal estimates.

  3. 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.