Paper · May 2, 2019

Small angle limits of Hamilton's footballs

Yanir A. Rubinstein, Kewei Zhang

math.DGarXiv:1905.00865
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Abstract

Compact Ricci solitons on surfaces have at most two cone points, and are known as Hamilton's footballs. In this note we completely describe the degenerations of these footballs as one or both of the cone angles approaches zero. In particular, we show that Hamilton's famous non-compact cigar soliton is the Gromov–Hausdorff limit of Hamilton's compact conical teardrop solitons.

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Submitted
May 2, 2019
Subjects
math.DG
Journal
Bull. London Math. Soc. 52 (2020) 189-199
DOI
10.1112/blms.12319
Ricci solitonsAncient solutionsSurfacesLimit spaces & RCDBoundaries & non-compact

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 Ricci iteration on homogeneous spaces (Artem Pulemotov and Yanir A. Rubinstein, 2016)2Pulemotov 2016 Metric contraction of the cone divisor by the conical Kähler-Ricci flow (Gregory Edwards, 2017)7Edwards 2017 Convergence of the Kähler-Ricci iteration (Tamás Darvas and Yanir A. Rubinstein, 2017)5Darvas 2017 On the Ricci iteration for homogeneous metrics on spheres and projective spaces (Timothy Buttsworth, Artem Pulemotov, Yanir A. Rubinstein et al., 2018)3Buttsworth 2018 Basis divisors and balanced metrics (Yanir A. Rubinstein, Gang Tian and Kewei Zhang, 2020)1Rubinstein 2020 On harmonic and biharmonic maps from gradient Ricci solitons (Volker Branding, 2022)6Branding 2022 Contractive transport maps from S² to nearly spherical surfaces with positive Ricci curvature (Jordan Serres, 2025)8Serres 2025 Liouville theorem on p-biharmonic map from gradient Ricci soliton (Xiangzhi Cao, 2026)4Cao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Basis divisors and balanced metrics

    Yanir A. Rubinstein, Gang Tian and Kewei Zhang

    Topics Ricci solitons · Same authors Yanir A. Rubinstein, Kewei Zhang

  2. Ricci iteration on homogeneous spaces

    Artem Pulemotov and Yanir A. Rubinstein

    Topics Ancient solutions, Limit spaces & RCD · Same author Yanir A. Rubinstein · Shared terms “Hausdorff”, “Gromov”

  3. On the Ricci iteration for homogeneous metrics on spheres and projective spaces

    Timothy Buttsworth, Artem Pulemotov, Yanir A. Rubinstein et al.

    Topics Ancient solutions · Same author Yanir A. Rubinstein · Shared terms “Hamilton”, “completely”

  4. Liouville theorem on p-biharmonic map from gradient Ricci soliton

    Xiangzhi Cao

    Topics Surfaces, Ancient solutions, Ricci solitons · Shared terms “cigar soliton”

  5. Convergence of the Kähler-Ricci iteration

    Tamás Darvas and Yanir A. Rubinstein

    Topics Surfaces · Same author Yanir A. Rubinstein

  6. On harmonic and biharmonic maps from gradient Ricci solitons

    Volker Branding

    Topics Surfaces, Ancient solutions, Ricci solitons · Shared terms “cigar soliton”

  7. Metric contraction of the cone divisor by the conical Kähler-Ricci flow

    Gregory Edwards

    Topics Surfaces, Limit spaces & RCD · Shared terms “cone angle”, “Hausdorff”, “Gromov”

  8. Contractive transport maps from to nearly spherical surfaces with positive Ricci curvature

    Jordan Serres

    Topics Surfaces, Limit spaces & RCD

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 solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

  1. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  2. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.