Paper · Jun 4, 2020

Steady gradient Kähler-Ricci solitons on crepant resolutions of Calabi-Yau cones

Ronan J. Conlon, Alix Deruelle

math.DGarXiv:2006.03100
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Abstract

We show that, up to the flow of the soliton vector field, there exists a unique complete steady gradient Kähler-Ricci soliton in every Kähler class of an equivariant crepant resolution of a Calabi-Yau cone converging at a polynomial rate to Cao's steady gradient Kähler-Ricci soliton on the cone.

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Submitted
Jun 4, 2020
Subjects
math.DG, math.AP

Comments: 92 pages

Ricci solitonsKähler–Ricci solitonsRicci-flat & Calabi–YauKähler geometry

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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 Expanding Kähler-Ricci solitons coming out of Kähler cones (Ronan J. Conlon and Alix Deruelle, 2016)5Conlon 2016 Steady Kähler-Ricci solitons on crepant resolutions of finite quotients of ℂⁿ (Olivier Biquard and Heather Macbeth, 2017)4Biquard 2017 On finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces (Charles Cifarelli, Ronan J. Conlon and Alix Deruelle, 2022)8Cifarelli 2022 A new complete two-dimensional shrinking gradient Kähler-Ricci soliton (Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al., 2022)3Bamler 2022 The asymptotic behavior of the steady gradient Kähler-Ricci soliton of the Taub-NUT type of Apostolov and Cifarelli (Daheng Min, 2024)6Min 2024 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)7Babu 2025 Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones (Ronan J. Conlon and Alix Deruelle, 2026)2Conlon 2026 Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton (Ronan J. Conlon and Alix Deruelle, 2026)1Conlon 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 the soliton vector field of a shrinking gradient Kähler-Ricci soliton

    Ronan J. Conlon and Alix Deruelle

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

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

    Ronan J. Conlon and Alix Deruelle

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

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

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

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

  4. Steady Kähler-Ricci solitons on crepant resolutions of finite quotients of

    Olivier Biquard and Heather Macbeth

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “crepant resolutions”, “equivariant crepant”

  5. Expanding Kähler-Ricci solitons coming out of Kähler cones

    Ronan J. Conlon and Alix Deruelle

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

  6. The asymptotic behavior of the steady gradient Kähler-Ricci soliton of the Taub-NUT type of Apostolov and Cifarelli

    Daheng Min

    Topics Kähler–Ricci solitons, Ricci-flat & Calabi–Yau, Kähler geometry · Shared terms “gradient Kähler”, “steady gradient”, “crepant resolutions”

  7. 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

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

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

    Charles Cifarelli, Ronan J. Conlon and Alix Deruelle

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

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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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.