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In this new version, we give an affirmative solution to a conjecture of Chengproposed in 1979 which asserts that the Bergman metric of a smoothly boundedstrongly pseudoconvex domain in $\mathbb{C}^n, n\geq 2,$ is K\"ahler-Einsteinif and only if the domain is biholomorphic to the ball. We establish versionsof various classical theorems that are used in the solution for Stein spaces.Among other things, we construct a hyperbolic metric over a Stein space withspherical boundary. We also prove the Q. K. Lu type uniformization theorem forStein spaces with isolated normal singularities.

Bergman-Einstein metrics, hyperbolic metrics and Stein spaces with spherical boundaries