Bergman representative coordinate, constant holomorphic curvature and a multidimensional generalization of Carath\'eodory's theorem

TL;DR

This paper extends Lu's theorem to certain pseudoconvex domains with incomplete Bergman metrics of constant holomorphic sectional curvature, characterizes such domains, and generalizes Carathéodory's theorem on biholomorphic extension.

Contribution

It introduces a multidimensional generalization of Carathéodory's theorem and characterizes domains biholomorphic to a ball minus a pluripolar set using Bergman kernel techniques.

Findings

Characterization of domains biholomorphic to a ball minus a pluripolar set.

Extension of Lu's theorem to incomplete Bergman metric domains.

Conditions for boundary of a biholomorphic ball to be a topological sphere.

Abstract

By using the Bergman representative coordinate and Calabi's diastasis, we extend a theorem of Lu to bounded pseudoconvex domains whose Bergman metric is incomplete with constant holomorphic sectional curvature. We characterize such domains that are biholomorphic to a ball possibly less a relatively closed pluripolar set. We also provide a multidimensional generalization of Carath\'eodory's theorem on the continuous extension of the biholomorphisms up to the closures. In particular, sufficient conditions are given, in terms of the Bergman kernel, for the boundary of a biholomorphic ball to be a topological sphere.