Paper · Jul 8, 2019

Classification of generalized Kähler-Ricci solitons on complex surfaces

Jeffrey Streets, Yury Ustinovskiy

math.DGarXiv:1907.03819
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Abstract

Using toric geometry we give an explicit construction of the compact steady solitons for pluriclosed flow first constructed in arXiv:1802.00170. This construction also reveals that these solitons are generalized Kähler in two distinct ways, with vanishing and nonvanishing Poisson structure. This gives the first examples of generalized Kähler structures with nonvanishing Poisson structure on non-standard Hopf surfaces, completing the existence question for such structures. Moreover this gives a complete answer to the existence question for generalized Kähler-Ricci solitons on compact complex surfaces. In the setting of generalized Kähler geometry with vanishing Poisson structure, we show that these solitons are unique. We show that these solitons are global attractors for the generalized Kähler-Ricci flow among metrics with maximal symmetry.

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Submitted
Jul 8, 2019
Subjects
math.DG, math.CV

Comments: 16 pages

Ricci flowKähler–Ricci flowGeneralized & G₂ flowsRicci solitonsKähler–Ricci solitonsKähler geometry

Its neighbourhood

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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 Generalized Kähler-Ricci flow and the classification of nondegenerate generalized Kähler surfaces (Jeffrey Streets, 2016)5Streets 2016 Deformation classes in generalized Kähler geometry (Matthew Gibson and Jeffrey Streets, 2020)7Gibson 2020 The Gibbons-Hawking ansatz in generalized Kähler geometry (Jeffrey Streets and Yury Ustinovskiy, 2020)4Streets 2020 Generalized Kähler-Ricci flow on toric Fano varieties (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2021)8Apostolov 2021 Variational structure and uniqueness of generalized Kähler-Ricci solitons (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2021)2Apostolov 2021 The generalized Kähler Calabi-Yau problem (Vestislav Apostolov, Xin Fu, Jeffrey Streets et al., 2022)3Apostolov 2022 Formal structure of scalar curvature in generalized Kähler geometry (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2024)6Apostolov 2024 Toric geometry of generalized Kähler-Ricci solitons (Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets et al., 2025)1Apostolov 2025
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  1. Toric geometry of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets et al.

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”, “toric geometry”

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

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”, “Hopf surfaces”

  3. The generalized Kähler Calabi-Yau problem

    Vestislav Apostolov, Xin Fu, Jeffrey Streets et al.

    Topics Generalized & G₂ flows, Kähler–Ricci flow, Kähler geometry · Same authors Jeffrey Streets, Yury Ustinovskiy · Shared terms “Kähler structures”

  4. The Gibbons-Hawking ansatz in generalized Kähler geometry

    Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”, “Poisson structure”

  5. Generalized Kähler-Ricci flow and the classification of nondegenerate generalized Kähler surfaces

    Jeffrey Streets

    Topics Kähler–Ricci flow, Kähler geometry · Same author Jeffrey Streets · Shared terms “generalized Kähler”, “Poisson structure”, “complex surfaces”

  6. Formal structure of scalar curvature in generalized Kähler geometry

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”

  7. Deformation classes in generalized Kähler geometry

    Matthew Gibson and Jeffrey Streets

    Topics Kähler–Ricci flow, Kähler geometry · Same author Jeffrey Streets · Shared terms “generalized Kähler”, “Poisson”, “Kähler structures”

  8. Generalized Kähler-Ricci flow on toric Fano varieties

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler–Ricci flow, Kähler geometry · Same authors Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized 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 flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) 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)

In the guidebook

  1. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

  2. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

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

  3. 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.