A note on exhaustion of hyperbolic complex manifolds

TL;DR

This paper studies how hyperbolic complex manifolds can be exhausted by pseudoconvex domains through sequences of holomorphic maps, focusing on boundary convergence behavior.

Contribution

It introduces a new perspective on exhaustion sequences in hyperbolic complex manifolds and analyzes boundary convergence properties.

Findings

Sequences of holomorphic maps can exhaust hyperbolic complex manifolds.

Boundary points can be approached via inverse images of a fixed point.

Provides conditions for convergence to boundary points in the exhaustion process.

Abstract

The purpose of this article is to investigate a hyperbolic complex manifold $M$ exhausted by a pseudoconvex domain $\Omega$ in $\mathbb C^n$ via an exhausting sequence $\{f_j\colon \Omega\to M\}$ such that $f_j^{-1}(a)$ converges to a boundary point $\xi_0 \in \partial \Omega$ for some point $a\in M$.