Paper · Jan 12, 2016

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

Jeffrey Streets

math.DGarXiv:1601.02981
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As on arXiv

Abstract

We study the generalized Kähler-Ricci flow on complex surfaces with nondegenerate Poisson structure, proving long time existence and convergence of the flow to a weak hyperKähler structure.

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Submitted
Jan 12, 2016
Subjects
math.DG, math.AP
Ricci flowKähler–Ricci flowRicci-flat & Calabi–YauKähler geometryLong-time existence & convergence

Its neighbourhood

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 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 Global viscosity solutions of generalized Kahler-Ricci flow (Jeffrey Streets, 2016)6Streets 2016 The nondegenerate generalized Kähler Calabi-Yau problem (Vestislav Apostolov and Jeffrey Streets, 2017)1Apostolov 2017 Classification of generalized Kähler-Ricci solitons on complex surfaces (Jeffrey Streets and Yury Ustinovskiy, 2019)2Streets 2019 Deformation classes in generalized Kähler geometry (Matthew Gibson and Jeffrey Streets, 2020)4Gibson 2020 Generalized Ricci Flow (Mario Garcia-Fernandez and Jeffrey Streets, 2020)8Garcia-Ferna 2020 The Gibbons-Hawking ansatz in generalized Kähler geometry (Jeffrey Streets and Yury Ustinovskiy, 2020)5Streets 2020 Generalized Kähler-Ricci flow on toric Fano varieties (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2021)7Apostolov 2021 The generalized Kähler Calabi-Yau problem (Vestislav Apostolov, Xin Fu, Jeffrey Streets et al., 2022)3Apostolov 2022
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The nondegenerate generalized Kähler Calabi-Yau problem

    Vestislav Apostolov and Jeffrey Streets

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, Long-time existence & convergence · Same author Jeffrey Streets · Shared terms “nondegenerate generalized”, “generalized Kähler”, “nondegenerate Poisson”

  2. Classification of generalized Kähler-Ricci solitons on complex surfaces

    Jeffrey Streets and Yury Ustinovskiy

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

  3. The generalized Kähler Calabi-Yau problem

    Vestislav Apostolov, Xin Fu, Jeffrey Streets et al.

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, Long-time existence & convergence · Same author Jeffrey Streets · Shared terms “generalized Kähler”, “hyperkähler”

  4. 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”

  5. The Gibbons-Hawking ansatz in generalized Kähler geometry

    Jeffrey Streets and Yury Ustinovskiy

    Topics Ricci-flat & Calabi–Yau, Kähler geometry · Same author Jeffrey Streets · Shared terms “generalized Kähler”, “Kähler surfaces”, “nondegenerate Poisson”

  6. Global viscosity solutions of generalized Kahler-Ricci flow

    Jeffrey Streets

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

  7. Generalized Kähler-Ricci flow on toric Fano varieties

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler–Ricci flow, Long-time existence & convergence, Kähler geometry · Same author Jeffrey Streets · Shared terms “generalized Kähler”

  8. Generalized Ricci Flow

    Mario Garcia-Fernandez and Jeffrey Streets

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, Kähler geometry · Same author Jeffrey Streets · 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) 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. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

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

  2. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.