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In this paper, we study the Bergman metric of a finite ball quotient$\mathbb{B}^n/\Gamma$, where $\Gamma \subseteq \mathrm{Aut}(\mathbb{B}^n)$ is afinite, fixed point free, abelian group. We prove that this metric isK\"ahler--Einstein if and only if $\Gamma$ is trivial, i.e., when the ballquotient $\mathbb{B}^n/\Gamma$ is the unit ball $\mathbb{B}^n$ itself. As aconsequence, we establish a characterization of the unit ball among normalStein spaces with isolated singularities and abelian fundamental groups interms of the existence of a Bergman-Einstein metric.

On the classification of normal Stein spaces and finite ball quotients with Bergman-Einstein metrics