Paper · Oct 6, 2016

Global viscosity solutions of generalized Kahler-Ricci flow

Jeffrey Streets

math.AParXiv:1610.01950
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Abstract

We apply ideas from viscosity theory to establish the existence of a unique global weak solution to the generalized Kahler-Ricci flow in the setting of commuting complex structures. Our results are restricted to the case of a smooth manifold with smooth background data. We discuss the possibility of extending these results to more singular settings, pointing out a key error in the existing literature on viscosity solutions to complex Monge-Ampere equations/Kahler-Ricci flow.

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Submitted
Oct 6, 2016
Subjects
math.AP, math.DG
Ricci flowKähler–Ricci flowKähler geometryWeak & singular flows

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)2Streets 2016 The nondegenerate generalized Kähler Calabi-Yau problem (Vestislav Apostolov and Jeffrey Streets, 2017)6Apostolov 2017 Classification of generalized Kähler-Ricci solitons on complex surfaces (Jeffrey Streets and Yury Ustinovskiy, 2019)3Streets 2019 Deformation classes in generalized Kähler geometry (Matthew Gibson and Jeffrey Streets, 2020)1Gibson 2020 Generalized Ricci Flow (Mario Garcia-Fernandez and Jeffrey Streets, 2020)7Garcia-Ferna 2020 Generalized Kähler-Ricci flow on toric Fano varieties (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2021)5Apostolov 2021 The generalized Kähler Calabi-Yau problem (Vestislav Apostolov, Xin Fu, Jeffrey Streets et al., 2022)4Apostolov 2022 Pluriclosed flow on Oeljeklaus-Toma manifolds (Jeffrey Streets and Xiaokang Wang, 2025)8Streets 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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”

  2. 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”

  3. 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”

  4. The generalized Kähler Calabi-Yau problem

    Vestislav Apostolov, Xin Fu, Jeffrey Streets et al.

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

  5. Generalized Kähler-Ricci flow on toric Fano varieties

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

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

  6. The nondegenerate generalized Kähler Calabi-Yau problem

    Vestislav Apostolov and Jeffrey Streets

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

  7. Generalized Ricci Flow

    Mario Garcia-Fernandez and Jeffrey Streets

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

  8. Pluriclosed flow on Oeljeklaus-Toma manifolds

    Jeffrey Streets and Xiaokang Wang

    Topics 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) 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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  3. 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.