Bergman-Einstein metrics, a generalization of Kerner's theorem and Stein spaces with spherical boundaries

We give an affirmative solution to a conjecture of Cheng proposed in 1979 which asserts that the Bergman metric of a smoothly bounded strongly pseudoconvex domain in C-n, n >= 2, is Kahler-Einstein if and only if the domain is biholomorphic to the ball. We establish a version of the classical Kerner theorem for Stein spaces with isolated singularities which has an immediate application to construct a hyperbolic metric over a Stein space with a spherical boundary.