A Note on the Complete Kähler–Einstein Metrics of Disk Bundles Over Compact Homogeneous Kähler Manifolds

In this article, we focus on the explicit description of the Kähler–Einstein metric on the disk bundle over some simply connected compact homogeneous Kähler manifolds. More precisely, we consider a strictly pseudoconvex domain in a Hermitian line bundle, which is the disk bundle of the γ -tensor power of the negative canonical bundle over any compact homogeneous Kähler manifold. We obtained a necessary and sufficient condition for the existence of the Kähler–Einstein metric on such disk bundle, which generalized a recent proposition by Ebenfelt, Xiao and Xu. As an application, we study the explicit solution of the Monge–Ampère equation on the disk bundles over the complex flag manifolds of classical type.