A uniformization theorem for the Bergman metric

Abstract

Let $D$ be a bounded domain in $\mathbb{C}^n$. Suppose the holomorphic sectional curvature of its Bergman metric equals a negative constant $τ$. We show that $D$ is biholomorphic to a domain $Ω$ equal to the unit ball in $\mathbb{C}^n$ less a relatively closed set of measure zero, and that all $L^2$-holomorphic functions on $Ω$ extend to $L^2$-holomorphic functions on the ball. Consequently, $τ$ must equal the holomorphic sectional curvature of the unit ball. This generalizes a classical theorem of Lu. Some applications of the theorem, especially in extending classical work of Wong and Rosay, are also presented.