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

Abstract

Let be a bounded domain in . Suppose the holomorphic sectional curvature of its Bergman metric equals a negative constant . We show that is biholomorphic to a domain equal to the unit ball in less a relatively closed set of measure zero, and that all ‐holomorphic functions on extend to ‐holomorphic functions on the ball. Consequently, 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.