Extending the Schwarz-Christoffel Formula to Universal Covering Maps of a Riemann Surface

Abstract

The classical Schwarz-Christoffel formula allows one to compute the conformal map of the unit disc onto a domain bounded by a polygon. An extension of this formula can be obtained that allows one to compute the conformal map of the unit disc onto a domain bounded by a “quadratic differential polygon” i.e. a closed Jordan curve that is the union of trajectory arcs of a certain quadratic differential and their end points. Now suppose that R is a finite Riemann surface with boundary and π: D → R is a universal covering map of R. Let Q(z) dz2 be a quadratic differential on R and let R be a sub-Riemann surface of R such that the boundary of R is the union of finitesided Q(z) dz2 polygons. If the lift of R, D = π- 1(R), is a simply-connected domain, then the Riemann mapping theorem ensures that there is a conformal map f of the unit disc onto D. In this paper, we will prove a generalization of the Schwarz-Christoffel formula that allows us to compute f. Then π ◦f is a universal covering map of R.