Paper · Jan 6, 2026

Stability of Hyperkähler Flow

Kuan-Hui Lee

math.DGWider flowsarXiv:2601.03092
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Abstract

In this work, we discuss the stability of Donaldson's flow of surfaces in a hyperkähler 4-manifold. In, Wang and Tsai proved a uniqueness theorem and dynamic stability theorem of the mean curvature flow for minimal surface. We extend their results and obtain a similar dynamic stability theorem of the hyperkähler flow.

Mean curvature flowRicci-flat & Calabi–Yau4-manifoldsKähler geometryStability & uniquenessMinimal & CMC surfaces

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

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

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

  4. HKT geometry on locally conformally hyperkähler manifolds

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

  5. Kähler stability of symplectic forms

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    Topics Ricci-flat & Calabi–Yau, Kähler geometry, Stability & uniqueness

  6. The parabolic quaternionic Calabi-Yau equation on hyperkähler manifolds

    Lucio Bedulli, Giovanni Gentili and Luigi Vezzoni PDF

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

  7. The Dirichlet problem for the minimal surface system on smooth domains

    Caiyan Li and Hengyu ZhouWider flows PDF

    Topics Minimal & CMC surfaces

  8. Classification of Minimal Surfaces and Solitons to the Mean Curvature Flow in as Translation Surfaces

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    Topics Minimal & CMC surfaces

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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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.