I would like to prove Schoenflies' theorem following C. Pommerenke, "Boundary Behaviour of Conformal Maps" (see Corollary 2.9, p. 25). In order to do that, I should first prove the previous statement, which is the following:
Corollary 2.8: A conformal map $f: \mathbb{D} \rightarrow G$ onto a Jordan domain $G \subseteq \mathbb{C}$ can be extended to a homeomorphism $f: \widehat{\mathbb{C}} \rightarrow \widehat{\mathbb{C}}$ such that $f(\infty) = \infty$ (Recall: $\widehat{C}: = \mathbb{C} \cup \{\infty\}$).
Given $f: \mathbb{D} \rightarrow G$ conformal, we may extend $f$ to a homeomorhism between the closures $f: \overline{D} \rightarrow \overline{G}$ using Caratheodory's theorem. Now, in the book we perform a mapping $f^{\ast}: \overline{D^{\ast}} \rightarrow \overline{{G}^{\ast}}$ between the closures of the outer domains, where $D^{\ast} : = \widehat{C} \setminus \overline{D}$ and ${G}^{\ast} : = \widehat{C} \setminus \overline{G}$.
I would like to explain how can I find such a conformal map $f^{\ast}$ between the outer domains using stereographic projection (which allows us to identify $\widehat{C}$ with $S^2 \subseteq \mathbb{R}^3$) and the Riemann mapping theorem (then the extension to a homeomorphism between the closures of the outer domains follows using again Caratheodory's theorem).
Any suggestions? Thanks in advance.