Paper · Nov 7, 2026

Uniqueness and stability of Appleton's solitons

Michael B. Law

math.DGarXiv:2611.10444
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Abstract

We prove that Appleton's noncollapsed steady gradient Ricci solitons on complex line bundles over positive Kähler–Einstein manifolds are the unique such solitons arising from his metric ansatz. We then show that the four-dimensional examples on , , are strictly linearly stable. By a result of Deruelle, this implies dynamical stability under the Ricci flow.

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Submitted
Nov 7, 2026
Subjects
math.DG

Comments: 56 pages. A Wolfram Mathematica notebook containing computational checks is included in the submission source

Ricci flowRicci solitonsEinstein metrics(Non)collapsing4-manifoldsKähler geometryStability & uniquenessKähler–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 A family of non-collapsed steady Ricci solitons in even dimensions greater or equal to four (Alexander Appleton, 2017)2Appleton 2017 Painlevé analysis of Ricci solitons over warped products (Alejandro Betancourt de la Parra, 2018)8de la Parra 2018 Compact Hermitian symmetric spaces, coadjoint orbits, and the dynamical stability of the Ricci flow (Stuart James Hall, Thomas Murphy and James Waldron, 2018)5Hall 2018 An optimal volume growth estimate for noncollapsed steady gradient Ricci solitons (Richard H. Bamler, Pak-Yeung Chan, Zilu Ma et al., 2021)7Bamler 2021 SU(2)-invariant steady gradient Ricci solitons on four-manifolds (Timothy Buttsworth, 2021)6Buttsworth 2021 Infinitely many non-collapsed steady Ricci solitons on complex line bundles (Hanci Chi, 2024)3Chi 2024 Ricci Flow on CP1-bundles over a Product of Kähler-Einstein Manifolds (Frederick Tsz-Ho Fong and Hung Tran, 2026)4Fong 2026 Classification of cohomogeneity one steady gradient Ricci solitons in dimension four and five (Yuxing Deng and Guohao Wang, 2026)1Deng 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 cohomogeneity one steady gradient Ricci solitons in dimension four and five

    Yuxing Deng and Guohao Wang

    Topics 4-manifolds, (Non)collapsing, Stability & uniqueness · Shared terms “Appleton”, “steady gradient”, “noncollapsed steady”

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

    Alexander Appleton

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

  3. Infinitely many non-collapsed steady Ricci solitons on complex line bundles

    Hanci Chi

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

  4. Ricci Flow on CP1-bundles over a Product of Kähler-Einstein Manifolds

    Frederick Tsz-Ho Fong and Hung Tran

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry

  5. Compact Hermitian symmetric spaces, coadjoint orbits, and the dynamical stability of the Ricci flow

    Stuart James Hall, Thomas Murphy and James Waldron

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry · Shared terms “dynamical stability”

  6. -invariant steady gradient Ricci solitons on four-manifolds

    Timothy Buttsworth

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

  7. An optimal volume growth estimate for noncollapsed steady gradient Ricci solitons

    Richard H. Bamler, Pak-Yeung Chan, Zilu Ma et al.

    Topics (Non)collapsing, Ricci solitons · Shared terms “Appleton solitons”, “noncollapsed steady”, “steady gradient”

  8. 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”

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) 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) 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. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.