The curvature estimation of the complete K\"ahler-Einstein metrics on the disk bundles

Abstract

In this paper, we study some geometric properties of the disk bundles over the complete K\"ahler manifolds. When the base space is the complete K\"ahler-Einstein manifold, we compute the holomorphic sectional curvature, Riemannian sectional curvature of the complete K\"ahler-Einstein metric on the disk bundle. Moreover, we present a parametric criteria for whether the manifold admits negative pinched property. When the base space is a bounded pseudoconvex domain equipped with its complete K\"ahler metric, the corresponding ball bundle in the $k$-direct sum of line bundles is a bounded pseudoconvex Hartogs domain with fiber dimension $k$ . If the base domain is simply-connected and $k\geq2$, we prove that the Bergman metric on such a Hartogs domain is K\"ahler-Einstein if and only if the domain is biholomorphically equivalent to a unit ball.