Estimates of the euclidean span for an open Riemann surface of genus one (Q1210054)

Every open Riemann surfaces of finite genus can be embedded into a compact Riemann surface of the same genus. M. Shiba proved that the moduli set of compact continuations of an open Riemann surface \(R\) of genus one is a closed disk (or a point) in the upper half plane. The euclidean (resp. noneuclidean) diameter of this disc is called the euclidean (resp. hyperbolic) span for \(R\). In this paper estimates for the Euclidean span for \(R\) are given for Riemann surfaces of the form \(R=\widetilde {R}/G\), where \(G\) is a group generated by two translations of \(\mathbb{C}\) and \(\widetilde {R}\) is a \(G\)-invariant domain of \(\mathbb{C}\). The case of a torus with horizontal slits is also studied.