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

TL;DR

This paper characterizes bounded domains with Bergman metrics of constant negative curvature, showing they are essentially the unit ball minus a measure-zero set, and explores implications for holomorphic function extension.

Contribution

It generalizes Lu's classical theorem by describing domains with constant curvature Bergman metrics as the unit ball minus a negligible set and examines related extension properties.

Findings

Domains with constant curvature Bergman metrics are biholomorphic to the unit ball minus a measure-zero set.

All L^2-holomorphic functions on such domains extend to the entire unit ball.

The curvature must match that of the unit ball, confirming a unique geometric characterization.

Abstract

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 $\Omega$ 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 $\Omega$ 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.