Paper · Aug 28, 2018

Pluriclosed flow and the geometrization of complex surfaces

Jeffrey Streets

math.DGarXiv:1808.09490
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Abstract

We recall fundamental aspects of the pluriclosed flow equation and survey various existence and convergence results, and the various analytic techniques used to establish them. Building on this, we formulate a precise conjectural description of the long time behavior of the flow on complex surfaces. This suggests an attendant geometrization conjecture which has implications for the topology of complex surfaces and the classification of generalized Kähler structures.

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Submitted
Aug 28, 2018
Subjects
math.DG, math.CV
Generalized & G₂ flowsGeometrizationKähler 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 Generalized Kähler-Ricci flow and the classification of nondegenerate generalized Kähler surfaces (Jeffrey Streets, 2016)7Streets 2016 Classification of generalized Kähler-Ricci solitons on complex surfaces (Jeffrey Streets and Yury Ustinovskiy, 2019)3Streets 2019 Generalized Ricci Flow (Mario Garcia-Fernandez and Jeffrey Streets, 2020)5Garcia-Ferna 2020 Ricci-Yang-Mills flow on surfaces and pluriclosed flow on elliptic fibrations (Jeffrey Streets, 2021)4Streets 2021 Geometrization in Geometry (Izabella Muraro de Freitas and Álvaro Krüger Ramos, 2022)8de Freitas 2022 The generalized Kähler Calabi-Yau problem (Vestislav Apostolov, Xin Fu, Jeffrey Streets et al., 2022)6Apostolov 2022 Pluriclosed flow and the Hull-Strominger system (Mario Garcia-Fernandez, Raul Gonzalez Molina and Jeffrey Streets, 2024)2Garcia-Ferna 2024 Pluriclosed flow on Oeljeklaus-Toma manifolds (Jeffrey Streets and Xiaokang Wang, 2025)1Streets 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Pluriclosed flow on Oeljeklaus-Toma manifolds

    Jeffrey Streets and Xiaokang Wang

    Topics Generalized & G₂ flows, Long-time existence & convergence, Kähler geometry · Same author Jeffrey Streets · Shared terms “pluriclosed flow”, “conjectural”

  2. Pluriclosed flow and the Hull-Strominger system

    Mario Garcia-Fernandez, Raul Gonzalez Molina and Jeffrey Streets

    Topics Geometrization, Generalized & G₂ flows, Long-time existence & convergence · Same author Jeffrey Streets · Shared terms “pluriclosed flow”, “conjectural”

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

    Jeffrey Streets and Yury Ustinovskiy

    Topics Generalized & G₂ flows, Kähler geometry · Same author Jeffrey Streets · Shared terms “complex surfaces”, “pluriclosed flow”, “Kähler structures”

  4. Ricci-Yang-Mills flow on surfaces and pluriclosed flow on elliptic fibrations

    Jeffrey Streets

    Topics Generalized & G₂ flows, Long-time existence & convergence · Same author Jeffrey Streets · Shared terms “pluriclosed flow”, “description”

  5. Generalized Ricci Flow

    Mario Garcia-Fernandez and Jeffrey Streets

    Topics Generalized & G₂ flows, Kähler geometry · Same author Jeffrey Streets · Shared terms “pluriclosed flow”, “fundamental aspects”

  6. The generalized Kähler Calabi-Yau problem

    Vestislav Apostolov, Xin Fu, Jeffrey Streets et al.

    Topics Generalized & G₂ flows, Long-time existence & convergence, Kähler geometry · Same author Jeffrey Streets · Shared terms “Kähler structures”, “formulate”

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

    Jeffrey Streets

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

  8. Geometrization in Geometry

    Izabella Muraro de Freitas and Álvaro Krüger Ramos

    Topics Geometrization

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.

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) Geometrization Thurston's conjecture: every closed 3-manifold can be cut along spheres and tori into pieces that each carry one of eight model geometries. Start with: Three dimensional manifolds, Kleinian groups and hyperbolic geometry (Thurston, 1982) 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  2. 10A.8
    Thurston’s Eight GeometriesBook 10A · Three-Manifolds and Geometrization

    The model geometries, the geometrization conjecture, and why it implies Poincaré.

  3. 10A.9
    Hyperbolic Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    Ideal tetrahedra, Mostow rigidity, and the thick–thin decomposition.