Paper · Aug 21, 2024

Pluriclosed flow and the Hull-Strominger system

Mario Garcia-Fernandez, Raul Gonzalez Molina, Jeffrey Streets

math.DGarXiv:2408.11674
PDF Abstract page

As on arXiv

Abstract

We define a natural extension of pluriclosed flow aiming at constructing solutions of the Hull-Strominger system. We give several geometric formulations of this flow, which yield a series of a priori estimates for the flow and also for the Hull-Strominger system. The evolution equations are derived using the theory of string algebroids, a class of Courant algebroids which occur naturally in higher gauge theory. Using this, we interpret the flow as generalized Ricci flow and also as a higher/coupled version of Hermitian-Yang-Mills flow, proving furthermore that it is compatible with symmetry reduction. Regarding our main analytical results, we prove a priori estimates for uniformly parabolic solutions. This in particular settles the question of smooth regularity of uniformly elliptic solutions of the Hull-Strominger system, generalizing Yau's estimate for the complex Monge-Ampère equation. We prove global existence and convergence results for the flow on special backgrounds, and discuss a conjectural relationship of the flow to the geometrization of Reid's fantasy.

Ricci flowGeneralized & 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 2021, 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 Four-dimensional generalized Ricci flows with nilpotent symmetry (Steven Gindi and Jeffrey Streets, 2021)7Gindi 2021 The generalized Kähler Calabi-Yau problem (Vestislav Apostolov, Xin Fu, Jeffrey Streets et al., 2022)4Apostolov 2022 Optimal Transport and Generalized Ricci Flow (Eva Kopfer and Jeffrey Streets, 2023)8Kopfer 2023 Ricci flow on Courant algebroids (Jeffrey Streets, Charles Strickland-Constable and Fridrich Valach, 2024)3Streets 2024 The Strominger System and Flows by the Ricci Tensor (Sébastien Picard, 2024)6Picard 2024 The canonical symmetry reduction of string backgrounds (Aaron Kennon and Jeffrey Streets, 2025)2Kennon 2025 Pluriclosed flow on Oeljeklaus-Toma manifolds (Jeffrey Streets and Xiaokang Wang, 2025)1Streets 2025 Cohomogeneity one solutions of the IIB system (Lorenzo Foscolo and Mario Garcia-Fernandez, 2026)5Foscolo 2026
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 PDF

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

  2. The canonical symmetry reduction of string backgrounds

    Aaron Kennon and Jeffrey Streets PDF

    Topics Generalized & G₂ flows · Same author Jeffrey Streets · Shared terms “Hull Strominger”, “Strominger system”, “symmetry reduction”

  3. Ricci flow on Courant algebroids

    Jeffrey Streets, Charles Strickland-Constable and Fridrich Valach PDF

    Topics Generalized & G₂ flows · Same author Jeffrey Streets · Shared terms “algebroids”, “Courant”, “constructing solutions”

  4. The generalized Kähler Calabi-Yau problem

    Vestislav Apostolov, Xin Fu, Jeffrey Streets et al. PDF

    Topics Generalized & G₂ flows, Long-time existence & convergence, Kähler geometry · Same author Jeffrey Streets · Shared terms “global existence”, “priori estimates”, “Yau”

  5. Cohomogeneity one solutions of the IIB system

    Lorenzo Foscolo and Mario Garcia-Fernandez PDF

    Topics Generalized & G₂ flows, Kähler geometry · Same author Mario Garcia-Fernandez · Shared terms “Strominger”, “pluriclosed”, “Hermitian”

  6. The Strominger System and Flows by the Ricci Tensor

    Sébastien Picard PDF

    Topics Kähler geometry · Shared terms “Strominger system”, “Yau”

  7. Four-dimensional generalized Ricci flows with nilpotent symmetry

    Steven Gindi and Jeffrey Streets PDF

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

  8. Optimal Transport and Generalized Ricci Flow

    Eva Kopfer and Jeffrey Streets PDF

    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) 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. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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.