Exhaustion of hyperbolic complex manifolds and relations to the squeezing function

Abstract: The purpose of this article is twofold. The first aim is to characterize an $n$-dimensional hyperbolic complex manifold $M$ exhausted by a sequence ${\Omega_j}$ of domains in $\mathbb C^n$ via an exhausting sequence ${f_j\colon \Omega_j\to M}$ such that $f_j^{-1}(a)$ converges to a boundary point $\xi_0 \in \partial \Omega$ for some point $a\in M$. Then, our second aim is to show that any spherically extreme boundary point must be strongly pseudoconvex.