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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.

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