A note on exhaustion of hyperbolic complex manifolds

Abstract

The purpose of this article is to investigate a hyperbolic complex manifold M M exhausted by a pseudoconvex domain Ω \Omega in C n \mathbb {C}^n via an exhausting sequence { f j : Ω → M } \{f_j\colon \Omega \to M\} such that f j − 1 ( a ) f_j^{-1}(a) converges to a boundary point ξ 0 ∈ ∂ Ω \xi _0 \in \partial \Omega for some point a ∈ M a\in M .