Paper · Nov 28, 2021

Symplectic geometric flows

Teng Fei, Duong H. Phong

math.SGarXiv:2111.14048
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Abstract

Several geometric flows on symplectic manifolds are introduced which are potentially of interest in symplectic geometry and topology. They are motivated by the Type IIA flow and T-duality between flows in symplectic geometry and flows in complex geometry. Examples include the Hitchin gradient flow on symplectic manifolds, and a new flow which is called the dual Ricci flow.

Comments: 16 pages, comments welcome!

Ricci flow

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 Kähler stability of symplectic forms (Jeffrey Streets and Gang Tian, 2022)5Streets 2022 Diameter estimates in Kähler geometry (Bin Guo, Duong H. Phong, Jian Song et al., 2022)2Guo 2022 Geometric flows from unified string theories (Duong H. Phong, 2023)1Phong 2023 A Bernstein theorem for ancient solution to symplectic mean curvature flow (Xiangzhi Cao, 2023)4Cao 2023 Local Existence Of The Symplectic Gradient Flow On The Hyperkähler Four-dimensional Flat Torus (Pinsard Morel Lucas, 2025)6Lucas 2025 Long-time existence of some geometric flows with bounded scalar curvature (Chuanhuan Li and Yi Li, 2026)7Li 2026 Regularizing estimates for positive solutions of the heat equation under geometric flows (Alessandro Goffi and Francesco Pediconi, 2026)8Goffi 2026 Kähler-Ricci Flow is Symplectic Deflation plus an Exact Symplectic Deformation (Randall R. Van Why, 2026)3Van Why 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Geometric flows from unified string theories

    Duong H. Phong PDF

    Topics Ricci flow · Same author Duong H. Phong · Shared terms “symplectic geometry”, “geometric flows”, “complex geometry”

  2. Diameter estimates in Kähler geometry

    Bin Guo, Duong H. Phong, Jian Song et al. PDF

    Topics Ricci flow · Same author Duong H. Phong

  3. Kähler-Ricci Flow is Symplectic Deflation plus an Exact Symplectic Deformation

    Randall R. Van Why PDF

    Topics Ricci flow · Shared terms “symplectic”

  4. A Bernstein theorem for ancient solution to symplectic mean curvature flow

    Xiangzhi CaoWider flows PDF

    Shared terms “symplectic”

  5. Kähler stability of symplectic forms

    Jeffrey Streets and Gang TianWider flows PDF

    Shared terms “symplectic”

  6. Local Existence Of The Symplectic Gradient Flow On The Hyperkähler Four-dimensional Flat Torus

    Pinsard Morel Lucas PDF

    Topics Ricci flow · Shared terms “symplectic”, “gradient flow”

  7. Long-time existence of some geometric flows with bounded scalar curvature

    Chuanhuan Li and Yi Li PDF

    Topics Ricci flow · Shared terms “geometric flows”

  8. Regularizing estimates for positive solutions of the heat equation under geometric flows

    Alessandro Goffi and Francesco Pediconi PDF

    Topics Ricci flow · Shared terms “geometric flows”

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)

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. 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.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.