Paper · Apr 23, 2025

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

Pinsard Morel Lucas

math.DGarXiv:2504.16494
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Abstract

Introducing a moment map whose zero locus is the group of symplectomorphisms of the real four-dimensional torus, we exhibit a gradient flow that can be made into a strictly parabolic flow by mean of a DeTurck trick (famously known for its use in the study of the Ricci flow), showing the local existence and regularity for the solutions of this flow and hence showing that the group of symplectomorphisms of the real four-dimensional torus is locally contractible. This work follows the ideas introduced by Yann Rollin in [3], even though the moment map picture comes from different considerations.

Ricci flowRicci-flat & Calabi–Yau4-manifoldsKähler geometryHarmonic maps & DeTurck trick

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  1. Scalar curvature rigidity of the four-dimensional sphere

    Simone Cecchini, Jinmin Wang, Zhizhang Xie et al. PDF

    Topics Harmonic maps & DeTurck trick, 4-manifolds · Shared terms “four dimensional”

  2. Harmonic spinors in the Ricci flow

    Julius Baldauf PDF

    Topics Ricci-flat & Calabi–Yau, 4-manifolds · Shared terms “gradient flow”

  3. Stability of Hyperkähler Flow

    Kuan-Hui LeeWider flows PDF

    Topics Ricci-flat & Calabi–Yau, 4-manifolds, Kähler geometry · Shared terms “hyperkähler”

  4. Integrable Deformations and Stability of the Ricci Flow

    Maxwell Stolarski and Alex Waldron PDF

    Topics Harmonic maps & DeTurck trick, Ricci-flat & Calabi–Yau

  5. On a dichotomy of the curvature decay of steady Ricci soliton

    Pak-Yeung Chan and Bo Zhu PDF

    Topics Ricci-flat & Calabi–Yau, 4-manifolds · Shared terms “four dimensional”

  6. Ricci Flow on ALF manifolds

    Dain Kim and Tristan Ozuch PDF

    Topics Ricci-flat & Calabi–Yau, Kähler geometry · Shared terms “hyperkähler”, “gradient flow”

  7. Ricci flow of -metrics in four dimensions

    Tobias Lamm and Miles Simon PDF

    Topics Harmonic maps & DeTurck trick, 4-manifolds

  8. Classification of Gradient Ricci solitons with harmonic Weyl curvature

    Jongsu Kim PDF

    Topics Ricci-flat & Calabi–Yau, 4-manifolds · Shared terms “four dimensional”

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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.

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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… 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)

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.