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 Kähler-Einstein manifold $(M,\omega ,J)$ such that either $c_1(J)<0$ or $(M,\omega ,J)$ is a Calabi-Yau manifold, then the Kähler-Einstein structure $(\omega ,J)$ is dynamically stable.

29 pages