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

TL;DR

This paper proves that the Bergman metric of a smoothly bounded strongly pseudoconvex domain in complex space is Kähler-Einstein if and only if the domain is biholomorphic to the ball, confirming a long-standing conjecture.

Contribution

It provides an affirmative solution to Cheng's conjecture and extends classical theorems to Stein spaces, including constructing hyperbolic metrics with spherical boundaries.

Findings

Bergman metric is Kähler-Einstein iff the domain is biholomorphic to the ball

Constructed hyperbolic metrics on Stein spaces with spherical boundary

Proved a uniformization theorem for Stein spaces with isolated singularities

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 $\mathbb{C}^n, n\geq 2,$ is K\"ahler-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.