A metric analogue of Hartogs' theorem

TL;DR

This paper establishes a metric analogue of Hartogs' theorem, showing that under certain conditions, the Kobayashi metric on a strongly pseudoconvex domain implies the universal cover is the unit ball.

Contribution

It introduces a metric version of Hartogs' theorem replacing holomorphic functions with Hermitian metrics and applies it to characterize universal covers of pseudoconvex domains.

Findings

Kobayashi metric being Kähler implies universal cover is the unit ball

Proves a metric analogue of Hartogs' theorem

Characterizes universal covers of certain domains

Abstract

In this paper we prove a metric version of Hartogs' theorem where the holomorphic function is replaced by a locally symmetric Hermitian metric. As an application, we prove that if the Kobayashi metric on a strongly pseudoconvex domain with $\mathcal{C}^2$ smooth boundary is a K\"ahler metric, then the universal cover of the domain is the unit ball.