Stability of the Almost Hermitian Curvature Flow

TL;DR

This paper investigates the stability of almost Hermitian structures under the Almost Hermitian Curvature flow, proving that certain Kähler-Einstein structures are dynamically stable on specific classes of manifolds.

Contribution

It establishes the dynamic stability of Kähler-Einstein structures under the flow on manifolds with negative first Chern class or Calabi-Yau structures.

Findings

Kähler-Einstein structures are dynamically stable on manifolds with $c_1(J)<0$.

Stability is proven for Calabi-Yau manifolds.

Flow solutions converge to fixed points near stable structures.

Abstract

The Almost Hermitian Curvature flow was introduced by Streets and Tian in order to study almost hermitian structures, with a particular interest in symplectic structures. This flow is given by a diffusion-reaction equation. Hence it is natural to ask the following: which almost hermitian structures are dynamically stable? An almost hermitian structure $(\omega,J)$ is dynamically stable if it is a fixed point of the flow and there exists a neighborhood $\mathcal{N}$ of $(\omega,J)$ such that for any almost hermitian structure $(\omega(0),J(0)) \in \mathcal{N}$ the solution of the Almost Hermitian Curvature flow starting at $(\omega(0),J(0))$ exists for all time and converges to a fixed point of the flow. We prove that on a closed K\"{a}hler-Einstein manifold $(M,\omega,J)$ such that either $c_1(J) <0$ or $(M,\omega,J)$ is a Calabi-Yau manifold, then the K\"{a}hler-Einstein structure $(\omega,J)$ is dynamically stable.