Bergman-Einstein metrics on two-dimensional Stein spaces

We show that the Bergman metric of the ball quotients ${\mathbb{B}}^2\mathrm{/}\mathrm{\Gamma }$, where $\mathrm{\Gamma }$ is a finite and fixed point free group, is Kähler-Einstein if and only if $\mathrm{\Gamma }$ is trivial. As a consequence, we characterize the unit ball ${\mathbb{B}}^2$, among 2 dimensional Stein spaces with isolated normal singularities, proving an algebraic version of Cheng’s conjecture for 2 dimensional Stein spaces.