Paper · Feb 18, 2021

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

Jeffrey Streets

math.DGarXiv:2102.09538v2
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Abstract

We give a complete description of the global existence and convergence for the Ricci-Yang-Mills flow on bundles over Riemann surfaces. These results equivalently describe solutions to generalized Ricci flow and pluriclosed flow with symmetry.

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Submitted
Feb 18, 2021
Last revised
Nov 8, 2021 (v2)
Subjects
math.DG, math.AP

Comments: final version, to appear in Adv. Math

Ricci flowGeneralized & G₂ flowsSurfacesLong-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 Pluriclosed flow and the geometrization of complex surfaces (Jeffrey Streets, 2018)3Streets 2018 Structure of collapsing solutions of generalized Ricci flow (Steven Gindi and Jeffrey Streets, 2018)8Gindi 2018 Classification of generalized Kähler-Ricci solitons on complex surfaces (Jeffrey Streets and Yury Ustinovskiy, 2019)6Streets 2019 Generalized Ricci Flow (Mario Garcia-Fernandez and Jeffrey Streets, 2020)4Garcia-Ferna 2020 Four-dimensional generalized Ricci flows with nilpotent symmetry (Steven Gindi and Jeffrey Streets, 2021)5Gindi 2021 Pluriclosed flow and the Hull-Strominger system (Mario Garcia-Fernandez, Raul Gonzalez Molina and Jeffrey Streets, 2024)2Garcia-Ferna 2024 The Cauchy problem for gradient generalized Ricci solitons on a bundle gerbe (Severin Bunk, Miguel Pino Carmona and C. S. Shahbazi, 2025)7Bunk 2025 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 · Same author Jeffrey Streets · Shared terms “pluriclosed flow”, “global existence”

  2. Pluriclosed flow and the Hull-Strominger system

    Mario Garcia-Fernandez, Raul Gonzalez Molina and Jeffrey Streets

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

  3. Pluriclosed flow and the geometrization of complex surfaces

    Jeffrey Streets

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

  4. Generalized Ricci Flow

    Mario Garcia-Fernandez and Jeffrey Streets

    Topics Generalized & G₂ flows · Same author Jeffrey Streets · Shared terms “pluriclosed flow”

  5. Four-dimensional generalized Ricci flows with nilpotent symmetry

    Steven Gindi and Jeffrey Streets

    Topics Generalized & G₂ flows, Long-time existence & convergence · Same author Jeffrey Streets

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

    Jeffrey Streets and Yury Ustinovskiy

    Topics Generalized & G₂ flows · Same author Jeffrey Streets · Shared terms “pluriclosed flow”

  7. The Cauchy problem for gradient generalized Ricci solitons on a bundle gerbe

    Severin Bunk, Miguel Pino Carmona and C. S. Shahbazi

    Topics Generalized & G₂ flows, Surfaces · Shared terms “Riemann surfaces”

  8. Structure of collapsing solutions of generalized Ricci flow

    Steven Gindi and Jeffrey Streets

    Topics Generalized & G₂ flows · Same author Jeffrey Streets

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) 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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) 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. 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.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  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.