Stability of the Almost Hermitian Curvature Flow

arXiv1308.6214MaRDI QIDQ6244383

D. J. Smith

Publication date: 28 August 2013

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 $mathcalN$ of $(omega,J)$ such that for any almost hermitian structure $(omega(0),J(0))inmathcalN$ 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.

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