Conformal mapping of geodesically slit tori and an application to the evaluation of the hyperbolic span (Q1327731)

In an earlier paper [Kodai Math. J. 16, 118-137 (1993; Zbl 0782.30036)] the first author studied mappings of an open Riemann surface of genus one (open torus) into a (closed) torus and introduced the concept of hyperbolic span. In this paper the authors study when a symmetric torus (one for which the period network is rectangular) with a (geodesic) slit in one of the fundamental directions can be mapped conformally onto another such with a slit in the other fundamental direction. This is readily seen to be equivalent to the problem of mapping a rectangle (with sides parallel to the axes), provided with a symmetrically placed horizontal slit onto another such with a symmetrically placed vertical slit so that the vertices of the rectangles correspond. Necessary and sufficient conditions are given in terms of relations involving complete elliptic integrals. An application is given to determine the hyperbolic span of a geodesically slit symmetric torus. Finally, some numerical examples are given.