Univalent harmonic mappings onto parallel slit domains (Q1063719)

Let D be a domain of arbitrary connectivity in \({\bar {\mathbb{C}}}\) containing \(\infty\) and \(a: D\to \{w:\) \(| w| <1\}\) an analytic function. The authors give a constructive description of an univalent solution f of the differential equation \(f_{\bar z}=\bar a\overline{f_ z}\) which maps D onto a horizontal slit domain and fulfills certain normalizing conditions. This theorem is valid under mild restrictions for D. Furthermore it is shown that f is the unique solution of this problem if D has only countable many boundary components.