Paper · Aug 28, 2018

Symplectic curvature flow revisited

Casey Lynn Kelleher

math.DGarXiv:1808.09510
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Abstract

We continue studying a parabolic flow of almost Kähler structures introduced by Streets and Tian which naturally extends Kähler-Ricci flow onto symplectic manifolds. In the system of primarily the symplectic form, almost complex structure, Chern torsion and Chern connection, we establish new formulas for the evolutions of canonical quantities, in particular those related to the Chern connection. Using this, we give an extended characterization of fixed points of the flow originally performed by Streets and Tian.

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Submitted
Aug 28, 2018
Subjects
math.DG, math.AP

Comments: 24 pages

Ricci flowKähler–Ricci flowOther curvature flowsKähler geometry

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 Kähler manifolds with negative holomorphic sectional curvature, Kähler-Ricci flow approach (Ryosuke Nomura, 2016)8Nomura 2016 An almost complex Chern-Ricci flow (Tao Zheng, 2017)2Zheng 2017 Scalar V-soliton equation and Kähler-Ricci flow on symplectic quotients (Chang Li, 2020)6Li 2020 Kähler stability of symplectic forms (Jeffrey Streets and Gang Tian, 2022)1Streets 2022 Geometric flows from unified string theories (Duong H. Phong, 2023)5Phong 2023 Static solutions to symplectic curvature flow in dimension four (Gavin Ball, 2023)7Ball 2023 Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds (Liding Huang and Xiaohua Zhu, 2026)4Huang 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. Kähler stability of symplectic forms

    Jeffrey Streets and Gang TianWider flows

    Topics Other curvature flows, Kähler geometry · Shared terms “symplectic curvature”, “symplectic form”

  2. An almost complex Chern-Ricci flow

    Tao Zheng

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “almost complex”, “Chern”, “complex structure”

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

    Randall R. Van Why

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “symplectic”

  4. Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds

    Liding Huang and Xiaohua Zhu

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “symplectic”, “Streets”, “extended”

  5. Geometric flows from unified string theories

    Duong H. Phong

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “symplectic”, “almost complex”

  6. Scalar V-soliton equation and Kähler-Ricci flow on symplectic quotients

    Chang Li

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “symplectic”, “Tian”

  7. Static solutions to symplectic curvature flow in dimension four

    Gavin BallWider flows

    Topics Other curvature flows, Kähler geometry · Shared terms “symplectic curvature”

  8. Kähler manifolds with negative holomorphic sectional curvature, Kähler-Ricci flow approach

    Ryosuke Nomura

    Topics Kähler–Ricci flow, Kähler geometry

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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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)

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. 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.

  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.