Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stability of Hyperkähler Flow

Kuan-Hui Lee

TL;DR

The paper studies the stability of Donaldson's hyperkähler flow (H-flow) for surfaces in hyperkähler 4-manifolds, extending dynamic stability results from mean curvature flow to this setting. It establishes the gradient-flow structure via the hyperkähler energy $E(f)=\int_S \lambda^2\,\rho$, derives evolution equations for the metric, volume form, and second fundamental form, and introduces the special H-flow constrained by $f^*(\theta)=\rho$ with $\theta=\overline{\omega}_2+i\overline{\omega}_3$. The central result shows that if a Lagrangian surface $\Gamma$ lies close to a compact, strongly stable complex Lagrangian $\Sigma$ (e.g., the zero section in the Eguchi–Hanson space), then the special H-flow exists for all time and converges smoothly to $\Sigma$, with convergence rate governed by $C^0$, $C^1$, and energy-type estimates. The analysis combines tubular-neighborhood geometry, precise control of the second fundamental form, and maximum-principle arguments to demonstrate long-time existence and smooth convergence, providing a robust stability framework for hyperkähler flows in dimension four.

Abstract

In this work, we discuss the stability of Donaldson's flow of surfaces in a hyperkähler 4-manifold. In \cite{WT2}, Wang and Tsai proved a uniqueness theorem and $C^1$ 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.

Stability of Hyperkähler Flow

TL;DR

The paper studies the stability of Donaldson's hyperkähler flow (H-flow) for surfaces in hyperkähler 4-manifolds, extending dynamic stability results from mean curvature flow to this setting. It establishes the gradient-flow structure via the hyperkähler energy , derives evolution equations for the metric, volume form, and second fundamental form, and introduces the special H-flow constrained by with . The central result shows that if a Lagrangian surface lies close to a compact, strongly stable complex Lagrangian (e.g., the zero section in the Eguchi–Hanson space), then the special H-flow exists for all time and converges smoothly to , with convergence rate governed by , , and energy-type estimates. The analysis combines tubular-neighborhood geometry, precise control of the second fundamental form, and maximum-principle arguments to demonstrate long-time existence and smooth convergence, providing a robust stability framework for hyperkähler flows in dimension four.

Abstract

In this work, we discuss the stability of Donaldson's flow of surfaces in a hyperkähler 4-manifold. In \cite{WT2}, 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.
Paper Structure (15 sections, 36 theorems, 181 equations)

This paper contains 15 sections, 36 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Hyperkähler Flow
  3. Introduction
  4. Hyperkähler Energy
  5. Evolution Formulas
  6. Special Hyperkähler Flow
  7. Strongly Stable Submanifolds
  8. Strongly stable conditions
  9. Eguchi-Hanson Space
  10. The stability of special Hyperkähler flow in strongly stable surface
  11. Tubular Neighborhood
  12. Previous Estimate
  13. $C^0$ and $C^1$ Estimate
  14. Long time existence
  15. Proof of Main Result

Key Result

Theorem 1.1

Let $f_t$ be a solution of the $H$-flow on $[0,T)$, for $0\leq T\leq \infty$. Suppose that $\lambda$ and $|\nabla^k\lambda|$ are uniformly bounded on $f_t(S)$ for any positive integer $k$. If there exists a constant $\alpha_0$ such that Then there exists $\alpha_k$ such that Also, if $T<\infty$, then

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Proposition 2.2: SW, Proposition 2.1
  • Proposition 2.3: SW, Proposition 2.2
  • Definition 2.4
  • Proposition 2.5
  • proof
  • ...and 54 more