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

Abstract

In this new version, 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 ${\mathbb{C}}^n,n\ge 2,$ is Kähler-Einstein if and only if the domain is biholomorphic to the ball. We establish versions of various classical theorems that are used in the solution for Stein spaces. Among other things, we construct a hyperbolic metric over a Stein space with spherical boundary. We also prove the Q. K. Lu type uniformization theorem for Stein spaces with isolated normal singularities.

significant update from last version

References in corpus (2)

• Uniformization of strictly pseudoconvex domains
• On plurisubharmonicity of the solution of the Fefferman equation and its applications to estimate the bottom of the spectrum of Laplace-Beltrami operators

Cited by in corpus (1)

• A Hall of Statistical Mirrors