Bergman-Einstein metrics on two-dimensional Stein spaces

TL;DR

This paper proves that the Bergman metric on certain two-dimensional Stein spaces is Kähler-Einstein only when the space is the unit ball, confirming a special case of Cheng's conjecture.

Contribution

It characterizes the unit ball among 2D Stein spaces with isolated singularities via the Kähler-Einstein property of the Bergman metric.

Findings

Bergman metric of ball quotients is Kähler-Einstein iff the quotient is trivial.

Characterization of the unit ball among 2D Stein spaces with singularities.

An algebraic version of Cheng's conjecture for 2D Stein spaces.

Abstract

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.