Extending the Schwarz-Christoffel formula to universal covering maps of a Riemann surface
Let \(Q(z)\,dz^2\) be a quadratic differential in a domain \(D^*\) in the Riemann sphere \(\hat{\mathbb{C}}\). A \(Q\)-polygon \(P\) is a closed Jordan curve in \(D^*\) such that \(\arg [Q(z)\,dz^2]\) is piecewise constant on \(P\). \textit{R. Kühnau} [J. Reine Angew. Math. 228, 93--132 (1967; Zbl 0176.37701)] discovered an extension of the Schwarz-Christoffel formula to a domain bounded by a \(Q\)-polygon. The author of the article under review derives an extension of the Schwarz-Christoffel formula to universal covering maps of a Riemann surface. Let \(R\) be a finite Riemann surface with boundary and \(\pi :\mathbb{D}\mapsto R\) be a universal covering map of \(R\). Let \(Q(z)\,dz^2\) be a quadratic differential on \(R\) and let \(\hat{R}\) be a sub-Riemann surface of \(R\) such that the boundary of \(\hat{R}\) is a union of finite-sided \(Q\)-polygons. If the lift of \(\hat{R}\), \(D=\pi^{-1}(\hat{R})\), is a simply connected domain, then by the Riemann mapping theorem there is a conformal map \(f\) of \(\mathbb{D}\) onto \(D\). The main result of the author is that \(f\) can be computed as a solution of a certain differential equation. Then \(\pi\circ f\) is a universal covering map of \(\hat{R}\).
scientific article; zbMATH DE number 6124809
scientific article