Negatively curved homogeneous almost Kähler Einstein manifolds with nonpositive curvature operator

Abstract

Given a homogeneous almost Kähler manifold $(M,J,g)$ with nonpositive curvature operator, we prove that if $g$ is an Einstein metric having negative sectional curvature, then the almost complex structure $J$ must be integrable. Furthermore, such $(M,J,g)$ eventually has constant negative holomorphic sectional curvature and hence is holomorphically isometric to a complex hyperbolic space.