Stability of the Almost Hermitian Curvature Flow

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.