Paper · Mar 24, 2019

Eguchi-Hanson singularities in U(2)-invariant Ricci flow

Alexander Appleton

math.DGarXiv:1903.09936
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Abstract

We show that a Ricci flow in four dimensions can develop singularities modeled on the Eguchi-Hanson space. In particular, we prove that starting from a class of asymptotically cylindrical -invariant initial metrics on , a Type II singularity modeled on the Eguchi-Hanson space develops in finite time. Furthermore, we show that for these Ricci flows the only possible blow-up limits are (i) the Eguchi-Hanson space, (ii) the flat orbifold, (iii) the 4d Bryant soliton quotiented by , and (iv) the shrinking cylinder . As a byproduct of our work, we also prove the existence of a new family of Type II singularities caused by the collapse of a two-sphere of self-intersection .

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Submitted
Mar 24, 2019
Subjects
math.DG
Ricci flowAncient solutionsSingularities4-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 2016, 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 A family of non-collapsed steady Ricci solitons in even dimensions greater or equal to four (Alexander Appleton, 2017)1Appleton 2017 On three-dimensional Type I κ-solutions to the Ricci flow (Yongjia Zhang, 2017)3Zhang 2017 The rigidity of S³×ℝ under ancient Ricci flow (Yongjia Zhang, 2017)2Zhang 2017 Ancient solutions to the Ricci flow in dimension 3 (S. Brendle, 2018)4Brendle 2018 Irregular Eguchi-Hanson type metrics and their soliton analogues (Akito Futaki, 2020)5Futaki 2020 Ricci flow with bounded curvature integrals (Shota Hamanaka, 2021)6Hamanaka 2021 On κ-solutions and canonical neighborhoods in 4d Ricci flow (Robert Haslhofer, 2023)7Haslhofer 2023 Blowdowns along Ricci flow in dimension four (Dain Kim and Tristan Ozuch, 2026)8Kim 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A family of non-collapsed steady Ricci solitons in even dimensions greater or equal to four

    Alexander Appleton

    Topics 4-manifolds, Ancient solutions · Same author Alexander Appleton · Shared terms “Bryant soliton”

  2. The rigidity of under ancient Ricci flow

    Yongjia Zhang

    Topics 4-manifolds, Ancient solutions, Singularities

  3. On three-dimensional Type I -solutions to the Ricci flow

    Yongjia Zhang

    Topics 4-manifolds, Ancient solutions, Singularities · Shared terms “shrinking cylinder”, “Bryant soliton”

  4. Ancient solutions to the Ricci flow in dimension 3

    S. Brendle

    Topics Ancient solutions, Singularities · Shared terms “shrinking cylinder”, “Bryant soliton”, “finite time”

  5. Irregular Eguchi-Hanson type metrics and their soliton analogues

    Akito Futaki

    Shared terms “Eguchi Hanson”

  6. Ricci flow with bounded curvature integrals

    Shota Hamanaka

    Topics 4-manifolds, Singularities · Shared terms “orbifold”, “finite time”

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

    Robert Haslhofer

    Topics 4-manifolds, Ancient solutions, Singularities · Shared terms “asymptotically cylindrical”

  8. Blowdowns along Ricci flow in dimension four

    Dain Kim and Tristan Ozuch

    Topics 4-manifolds, Singularities · Shared terms “self intersection”, “singularity modeled”, “initial metrics”

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 Singularities guide: Appleton: Type II singularities modelled on the Eguchi–Hanson space.

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) 4-manifolds Four is the first dimension where the singularities of Ricci flow are not classified. It is also the dimension of the still-open smooth Poincaré… Start with: The topology of four-dimensional manifolds (Freedman, 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.