The curvature estimation of the complete Kähler-Einstein metrics on the disk bundles

In this paper, we compute the holomorphic sectional curvature and Riemannian sectional curvature of the complete Kähler-Einstein metric on the disk bundle over any complete Kähler-Einstein manifold. Then we study whether it is negatively pinched. When the base space is a bounded pseudoconvex domain equipped with its complete Kähler-Einstein metric, the corresponding disk bundle is a pseudoconvex Hartogs domain. We prove that the Bergman metric on such a Hartogs domain is Kähler-Einstein if and only if the domain is biholomorphically equivalent to a unit ball.