Paper · Oct 1, 2026

Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton

Ronan J. Conlon, Alix Deruelle

math.DGarXiv:2610.02164
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Abstract

We establish the uniqueness of the soliton vector field of a complete shrinking gradient Kähler-Ricci soliton up to pushforward by biholomorphism.

Comments: 12 pages

Ricci solitonsKähler–Ricci solitonsKähler geometryStability & uniqueness

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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 On finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces (Charles Cifarelli, Ronan J. Conlon and Alix Deruelle, 2022)3Cifarelli 2022 An Aubin continuity path for shrinking gradient Kähler-Ricci solitons (Charles Cifarelli, Ronan J. Conlon and Alix Deruelle, 2022)5Cifarelli 2022 A new complete two-dimensional shrinking gradient Kähler-Ricci soliton (Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al., 2022)2Bamler 2022 Uniqueness of asymptotically conical shrinking gradient Kähler-Ricci solitons (Carlos Esparza, 2025)6Esparza 2025 An Aubin continuity path for asymptotically conical toric shrinking gradient Kähler-Ricci solitons: openness and a solution for t=0 (Ivin Babu, Ronan J. Conlon and Alix Deruelle, 2025)4Babu 2025 A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity (Richard H. Bamler, Eric Chen and Ronan J. Conlon, 2026)8Bamler 2026 Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones (Ronan J. Conlon and Alix Deruelle, 2026)1Conlon 2026 Rigidity of nonsteady gradient Kähler-Ricci solitons with constant scalar curvature (Matheus Andrade Ribeiro de Moura Horácio, 2026)7Horácio 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones

    Ronan J. Conlon and Alix Deruelle PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness · Same authors Ronan J. Conlon, Alix Deruelle · Shared terms “shrinking gradient”, “gradient Kähler”, “complete shrinking”

  2. A new complete two-dimensional shrinking gradient Kähler-Ricci soliton

    Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al. PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Ronan J. Conlon, Alix Deruelle · Shared terms “shrinking gradient”, “gradient Kähler”, “complete shrinking”

  3. On finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces

    Charles Cifarelli, Ronan J. Conlon and Alix Deruelle PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Ronan J. Conlon, Alix Deruelle · Shared terms “soliton vector”, “shrinking gradient”, “gradient Kähler”

  4. An Aubin continuity path for asymptotically conical toric shrinking gradient Kähler-Ricci solitons: openness and a solution for

    Ivin Babu, Ronan J. Conlon and Alix Deruelle PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Ronan J. Conlon, Alix Deruelle · Shared terms “shrinking gradient”, “gradient Kähler”

  5. An Aubin continuity path for shrinking gradient Kähler-Ricci solitons

    Charles Cifarelli, Ronan J. Conlon and Alix Deruelle PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Ronan J. Conlon, Alix Deruelle · Shared terms “shrinking gradient”, “gradient Kähler”

  6. Uniqueness of asymptotically conical shrinking gradient Kähler-Ricci solitons

    Carlos Esparza PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness · Shared terms “shrinking gradient”, “soliton vector”, “gradient Kähler”

  7. Rigidity of nonsteady gradient Kähler-Ricci solitons with constant scalar curvature

    Matheus Andrade Ribeiro de Moura Horácio PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness · Shared terms “soliton vector”, “gradient Kähler”, “vector field”

  8. A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity

    Richard H. Bamler, Eric Chen and Ronan J. Conlon PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same author Ronan J. Conlon · Shared terms “gradient Kähler”

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) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  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.