Paper · Aug 1, 2017

A family of non-collapsed steady Ricci solitons in even dimensions greater or equal to four

Alexander Appleton

math.DGarXiv:1708.00161v5
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Abstract

We construct a family of non-collapsed, non-Kähler, non-Einstein steady Ricci solitons in even dimensions greater or equal to four. These solitons exist on complex line bundles over Kähler-Einstein manifolds of positive scalar curvature. They include a four-dimensional -invariant, non-collapsed Riemannian steady soliton on each of the line bundles , of . Finally, we find Taub-Nut like Ricci solitons and demonstrate a new proof for the existence of the Bryant soliton.

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Submitted
Aug 1, 2017
Last revised
May 5, 2022 (v5)
Subjects
math.DG

Comments: Fixed gap in proof of Lemma 6.7 of previous version; general editing and additional explanations

Ricci solitonsAncient solutionsEinstein metrics(Non)collapsing4-manifoldsKähler geometryScalar curvatureKähler–Einstein & K-stability

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 Ancient Ricci Flow Solutions on Bundles (Peng Lu and Y. K. Wang, 2016)5Lu 2016 Painlevé analysis of Ricci solitons over warped products (Alejandro Betancourt de la Parra, 2018)3de la Parra 2018 Eguchi-Hanson singularities in U(2)-invariant Ricci flow (Alexander Appleton, 2019)4Appleton 2019 Curvature estimates for steady Ricci solitons (Pak-Yeung Chan, 2019)8Chan 2019 Infinitely many non-collapsed steady Ricci solitons on complex line bundles (Hanci Chi, 2024)1Chi 2024 Asymptotic Geometry of Four-Dimensional Steady Solitons (Aprameya Girish Hebbar and Natasa Sesum, 2026)6Hebbar 2026 O(3)-symmetry of four-dimensional steady Ricci solitons (Hanbing Fang, 2026)7Fang 2026 Uniqueness and stability of Appleton's solitons (Michael B. Law, 2026)2Law 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Infinitely many non-collapsed steady Ricci solitons on complex line bundles

    Hanci Chi

    Topics Kähler–Einstein & K-stability, (Non)collapsing, Einstein metrics · Shared terms “collapsed steady”, “steady Ricci”, “complex line”

  2. Uniqueness and stability of Appleton's solitons

    Michael B. Law

    Topics 4-manifolds, Kähler–Einstein & K-stability, (Non)collapsing · Shared terms “complex line”, “line bundle”, “four dimensional”

  3. Painlevé analysis of Ricci solitons over warped products

    Alejandro Betancourt de la Parra

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry · Shared terms “complex line”, “line bundle”, “steady Ricci”

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

    Alexander Appleton

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

  5. Ancient Ricci Flow Solutions on Bundles

    Peng Lu and Y. K. Wang

    Topics Kähler–Einstein & K-stability, (Non)collapsing, Einstein metrics · Shared terms “collapsed”

  6. Asymptotic Geometry of Four-Dimensional Steady Solitons

    Aprameya Girish Hebbar and Natasa Sesum

    Topics 4-manifolds, Ancient solutions, Scalar curvature · Shared terms “steady soliton”, “four dimensional”, “Bryant soliton”

  7. -symmetry of four-dimensional steady Ricci solitons

    Hanbing Fang

    Topics 4-manifolds, (Non)collapsing, Ricci solitons · Shared terms “steady Ricci”, “four dimensional”

  8. Curvature estimates for steady Ricci solitons

    Pak-Yeung Chan

    Topics 4-manifolds, Scalar curvature, Ricci solitons · Shared terms “steady Ricci”, “four dimensional”

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) (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.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.

  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.