$\def\abs#1{\left|#1\right|}$I would like to understand this example:
- Why is the following set a hyperbolic manifold?
$X=\{[1:z:w]\in \mathbb{CP}_2\mid0<\abs z< 1, \abs w < \abs{\exp(1/z)}\}$
It's an examples given in the book Hyperbolic Manifolds and Holomorphic Mappings: An Introduction by Kobayashi, in order to give a counterexample of an optimistic generalization of the Big Picard Theorem. They claim that it is biholomorphic to $\mathbb{D}\times\mathbb{D}^*$. I dont understand why.