Domains with Bergman metrics of constant curvature and Bergman-negligible subsets

Let $D$ be a bounded domain in ${\mathbb{C}}^n$. Suppose the holomorphic sectional curvature of its Bergman metric equals a negative constant $\tau$. We show that $D$ is biholomorphic to a domain $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ extend to $L^2$-holomorphic functions on the ball. Consequently, $\tau$ 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.