Paper · Dec 19, 2025

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

Ivin Babu, Ronan J. Conlon, Alix Deruelle

math.DGarXiv:2512.18137
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Abstract

We show that any toric asymptotically conical shrinking gradient Kähler-Ricci soliton on an anti-canonically polarised resolution of a Kähler cone satisfies a complex Monge-Ampère equation. We then set up an Aubin continuity path to solve the resulting equation and show that it has a solution at the initial value of the path parameter in the toric case. This we do by implementing another continuity method. Finally, we prove openness of the initial value of the path parameter independent of the toricity.

Comments: 67 pages

Ricci solitonsKähler–Ricci solitonsKähler geometryBoundaries & non-compact

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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)5Cifarelli 2022 An Aubin continuity path for shrinking gradient Kähler-Ricci solitons (Charles Cifarelli, Ronan J. Conlon and Alix Deruelle, 2022)1Cifarelli 2022 A new complete two-dimensional shrinking gradient Kähler-Ricci soliton (Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al., 2022)4Bamler 2022 Uniqueness of asymptotically conical shrinking gradient Kähler-Ricci solitons (Carlos Esparza, 2025)7Esparza 2025 A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity (Richard H. Bamler, Eric Chen and Ronan J. Conlon, 2026)6Bamler 2026 Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones (Ronan J. Conlon and Alix Deruelle, 2026)2Conlon 2026 Linear Stability of Steady and Expanding Kähler-Ricci Solitons (Lucas Lavoyer and Adam Thompson, 2026)8Lavoyer 2026 Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton (Ronan J. Conlon and Alix Deruelle, 2026)3Conlon 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 “Aubin continuity”, “continuity path”, “shrinking gradient”

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

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

  4. 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”

  5. 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”

  6. 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 “gradient Kähler”, “asymptotically conical”, “Kähler cone”

  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”, “conical shrinking”

  8. Linear Stability of Steady and Expanding Kähler-Ricci Solitons

    Lucas Lavoyer and Adam Thompson PDF

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

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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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.