Paper · Sep 29, 2023

Static solutions to symplectic curvature flow in dimension four

Gavin Ball

math.DGWider flowsarXiv:2310.00143v2
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Abstract

This article studies special solutions to symplectic curvature flow in dimension four. Firstly, we derive a local normal form for static solutions in terms of holomorphic data and use this normal form to show that every complete static solution to symplectic curvature flow in dimension four is Kahler-Einstein. Secondly, we perform an exterior differential systems analysis of the soliton equation for symplectic curvature flow and use the Cartan-Kahler theorem to prove a local existence and generality theorem for solitons.

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Submitted
Sep 29, 2023
Last revised
Oct 8, 2023 (v2)
Subjects
math.DG, math.SG

Comments: 9 pages. v2: fixed typos, corrected Cartan characters in Thm 4.1, small changes to exposition

Other curvature flowsEinstein metrics4-manifoldsKähler geometryKähler–Einstein & K-stability

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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)1Streets 2022 Hermitian Calabi functional in complexified orbits (Jie He and Kai Zheng, 2023)5He 2023 New expanding Ricci solitons starting in dimension four (Jan Nienhaus and Matthias Wink, 2023)7Nienhaus 2023 Topological Obstructions to the Existence of Compact Shrinking Ricci Solitons in Dimension Four (Cameron MacMahon, 2024)6MacMahon 2024 Geometry and Analysis of Gradient Ricci Solitons in Dimension Four (Xiaodong Cao and Hung Tran, 2024)3Cao 2024 Ricci Flow on CP1-bundles over a Product of Kähler-Einstein Manifolds (Frederick Tsz-Ho Fong and Hung Tran, 2026)4Fong 2026 Prescribed T-curvature flow on the four-dimensional unit ball (Pak Tung Ho, Cheikh Birahim Ndiaye, Liming Sun et al., 2026)2Ho 2026 Semistability and asymptotics of geometric flows (Shing Tak Lam, 2026)8Lam 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”

  2. Prescribed -curvature flow on the four-dimensional unit ball

    Pak Tung Ho, Cheikh Birahim Ndiaye, Liming Sun et al.Wider flows

    Topics Kähler–Einstein & K-stability, 4-manifolds, Other curvature flows

  3. Geometry and Analysis of Gradient Ricci Solitons in Dimension Four

    Xiaodong Cao and Hung Tran

    Topics 4-manifolds · Shared terms “dimension four”

  4. Ricci Flow on CP1-bundles over a Product of Kähler-Einstein Manifolds

    Frederick Tsz-Ho Fong and Hung Tran

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry

  5. Hermitian Calabi functional in complexified orbits

    Jie He and Kai ZhengWider flows

    Topics Kähler–Einstein & K-stability, Other curvature flows, Kähler geometry · Shared terms “symplectic”

  6. Topological Obstructions to the Existence of Compact Shrinking Ricci Solitons in Dimension Four

    Cameron MacMahon

    Topics 4-manifolds, Kähler geometry · Shared terms “dimension four”

  7. New expanding Ricci solitons starting in dimension four

    Jan Nienhaus and Matthias Wink

    Topics 4-manifolds, Einstein metrics · Shared terms “dimension four”

  8. Semistability and asymptotics of geometric flows

    Shing Tak LamWider flows

    Topics Kähler–Einstein & K-stability, Other curvature flows, Kähler geometry · Shared terms “holomorphic”

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 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)