On the Classification of Normal Stein Spaces and Finite Ball Quotients With Bergman–Einstein Metrics

Abstract

Abstract We study the Bergman metric of a finite ball quotient $\mathbb{B}^n/\Gamma $, where $n \geq 2$ and $\Gamma \subseteq{\operatorname{Aut}}({\mathbb{B}}^n)$ is a finite, fixed point free, abelian group. We prove that this metric is Kähler–Einstein if and only if $\Gamma $ is trivial, that is, when the ball quotient $\mathbb{B}^n/\Gamma $ is the unit ball ${\mathbb{B}}^n$ itself. As a consequence, we characterize the unit ball among normal Stein spaces with isolated singularities and abelian fundamental groups in terms of the existence of a Bergman–Einstein metric.