Paper · Jun 2, 2022

On geometry of steady toric Kähler-Ricci solitons

Yury Ustinovskiy

math.DGarXiv:2206.01196v2
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Abstract

In this paper we study the gradient steady Kähler-Ricci soliton metrics on non-compact toric manifolds. We show that the orbit space of the free locus of such a manifold carries a natural Hessian structure with a nonnegative Bakry-Émery tensor. We generalize Calabi's classical rigidity result and use this to prove that any complete -invariant gradient steady Kähler-Ricci soliton with a free torus action must be a flat .

Comments: 20 pages, minor corrections; new references added

Ricci solitonsKähler–Ricci solitonsKähler geometryLimit spaces & RCDStability & uniquenessBoundaries & non-compact

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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 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 Variational structure and uniqueness of generalized Kähler-Ricci solitons (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2021)2Apostolov 2021 Hamiltonian 2-forms and new explicit Calabi–Yau metrics and gradient steady Kähler–Ricci solitons on ℂⁿ (Vestislav Apostolov and Charles Cifarelli, 2023)5Apostolov 2023 Formal structure of scalar curvature in generalized Kähler geometry (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2024)3Apostolov 2024 Uniqueness of Kähler Ricci shrinkers on toric orbifolds (Yu Li and Wenjia Zhang, 2024)8Li 2024 Liouville theorems for harmonic 1-forms on gradient Ricci solitons (Chenghong He, Di Wu and Xi Zhang, 2024)6He 2024 Toric geometry of generalized Kähler-Ricci solitons (Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets et al., 2025)1Apostolov 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)7Babu 2025 Linear Stability of Steady and Expanding Kähler-Ricci Solitons (Lucas Lavoyer and Adam Thompson, 2026)4Lavoyer 2026
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  1. Toric geometry of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets et al. PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same author Yury Ustinovskiy · Shared terms “steady Kähler”

  2. Variational structure and uniqueness of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness · Same author Yury Ustinovskiy

  3. Formal structure of scalar curvature in generalized Kähler geometry

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness · Same author Yury Ustinovskiy · Shared terms “Calabi”

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

    Lucas Lavoyer and Adam Thompson PDF

    Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry

  5. Hamiltonian -forms and new explicit Calabi–Yau metrics and gradient steady Kähler–Ricci solitons on

    Vestislav Apostolov and Charles Cifarelli PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “steady Kähler”, “gradient steady”, “Calabi”

  6. Liouville theorems for harmonic 1-forms on gradient Ricci solitons

    Chenghong He, Di Wu and Xi Zhang PDF

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

  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 PDF

    Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry

  8. Uniqueness of Kähler Ricci shrinkers on toric orbifolds

    Yu Li and Wenjia Zhang PDF

    Topics Kähler geometry, Stability & uniqueness, Ricci solitons · Shared terms “toric manifolds”

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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.