On the distortion of relative circle domain isomorphisms (Q1385407)

A disk packing is a collection of closed disks with disjoint interiors in the Riemann sphere. Let \(P\) be a finite disk packing contained in a finitely connected domain \(\Omega\) in the sphere. The authors have previously introduced the notion of a finitely connected ``relative circle domain'' in \(\Omega: A=\Omega- \bigcup_{D\in P}\text{interior}(D)\). If \(P^*\) is another finite disk packing in a domain \(\Omega^*\), and if \(f: A\to A^*= \Omega^*- \bigcup_{D\in P^*}\text{interior}(D)\) is a conformal homeomorphism such that \(\partial\Omega\) corresponds to \(\partial\Omega^*\) under \(f\), then \(f\) is called a relative circle domain (rcd) isomorphism from \(\Omega\) to \(\Omega^*\). The authors construct counterexamples for certain properties of rcd isomorphisms which would be analogous to properties of conformal mappings. In particular, counterexamples to Koebe's 1/4 Theorem are constructed for rcd isomorphisms and for disk packings. A conformal mapping between two annuli must preserve the ``modulus'' of the annuli. The authors show by example that this is not true for rcd isomorphisms. Some positive results are obtained if the annuli are suitably restricted.