The inverse Riemann mapping theorem for relative circle domains (Q1918391)

A subdomain \(\Omega\) contained in a domain \(A\) in the Riemann sphere is called a relative circle domain in \(A\) if each component of \(A-\Omega\) is either a closed disk or a point. The authors have previously proved a Riemann mapping theorem for relative circle domains: Let \(A\) be a simply connected domain, not the whole plane, and \(\Omega\) a relative circle domain in \(A\). Then there exists a relative circle domain \(\Omega^*\) in the unit disk \(U\) and a conformal homeomorphism \(f\) of \(\Omega\) onto \(\Omega^*\), which extends to a conformal homeomorphism of \(A\) onto \(U\), whose complex dilatation vanishes a.e. in \(\partial\Omega\cap A\). Here, the authors prove the following inverse theorem: Let \(\Omega\) be a relative circle domain in the unit disk \(U\) and let \(A\) be a simply connected domain which is not the whole plane. Then for any \(z_0\in\Omega\) and \(z^*_0\in A\) there exists a relative circle domain \(\Omega^*\in A\) and a conformal homeomorphism \(f\) of \(\Omega\) onto \(\Omega^*\) which extends to a locally quasiconformal homeomorphism of \(U\) onto \(A\), whose complex dilatation vanishes a.e. in \(\partial\Omega\cap U\) and such that \(f(z_0)= z^*_0\), \(f'(z_0)>0\). A similar theorem for circle packings is stated.