On the Classification of Stein spaces with Bergman-Einstein metrics

For every $N\ge 2$, we prove that the Bergman metric on the regular locus of a finite ball quotient ${\mathbb{B}}^N\mathrm{/}\mathrm{\Gamma }$, where $\mathrm{\Gamma }\subset \mathrm{U}(N)$ is finite and fixed-point-free, is Kähler-Einstein if and only if $\mathrm{\Gamma }$ is trivial. Consequently, if $\mathrm{\Omega }$ is an $N$-dimensional normal Stein space with isolated singularities and compact, smooth, strongly pseudoconvex boundary admitting a real-algebraic CR realization, then the Bergman metric on ${\mathrm{\Omega }}_{\mathrm{r}\mathrm{e}\mathrm{g}}$ is Kähler-Einstein if and only if $\mathrm{\Omega }$ is biholomorphic to ${\mathbb{B}}^N$. This proves an algebraic version of the Cheng-Huang-Xiao conjecture in every complex dimension $N\ge 2$.