We show that the Bergman metric of the ball quotients $\mathbb{ B}^2/\Gamma$, where $\Gamma$ is a finite and fixed point free group, is K\"ahler-Einstein if and only if $\Gamma$ 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.
Categories: math.CV math.MG