Paper · Jul 28, 2026

Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones

Ronan J. Conlon, Alix Deruelle

math.DGarXiv:2607.26054
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Abstract

We show that any complete shrinking gradient Kähler-Ricci soliton on a resolution of a Kähler cone is necessarily asymptotically conical. From a result of Esparza, it then follows that up to pullback by biholomorphism, there exists at most one complete shrinking gradient Kähler-Ricci soliton on such a resolution. This confirms a special case of the uniqueness part of a conjecture of Song-Zhang. Some other consequences are also discussed.

Comments: 29 pages

Ricci solitonsKähler–Ricci solitonsKähler geometryStability & uniquenessBoundaries & non-compact

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

    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. 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, Boundaries & non-compact, Kähler geometry · Same authors Ronan J. Conlon, Alix Deruelle · Shared terms “shrinking gradient”, “gradient Kähler”, “Kähler cone”

  3. 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”

  4. 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, Boundaries & non-compact, Kähler geometry · Same author Ronan J. Conlon · Shared terms “Kähler cone”, “gradient Kähler”, “asymptotically conical”

  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. 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, Boundaries & non-compact, Kähler geometry · Same authors Ronan J. Conlon, Alix Deruelle · Shared terms “shrinking gradient”, “gradient Kähler”

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

    Carlos Esparza PDF

    Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry · Shared terms “shrinking gradient”, “gradient Kähler”, “biholomorphism”

  8. Shrinking gradient Kähler-Ricci solitons are simply-connected

    Carlos Esparza PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “shrinking gradient”, “gradient Kähler”, “shrinking 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.