On the moduli set of continuations of an open Riemann surface of genus one (Q1326650)

Let \(R\) be an open Riemann surface of genus 1 and \(\chi\) a canonical homology basis of \(R\) modulo dividing cycles. Let \(M(R,\chi)\) be the set of moduli of all marked tori into which \((R,\chi)\) can be conformally embedded. Shiba showed, that \(M(R,\chi)\) is a closed disk or a single point in the upper half plane. In this paper the moduli disk \(M(R,\chi)\) is characterized in terms of the extremal lengths of three families of curves on \(R\). The hyperbolic diameter of \(M(R,\chi)\) is calculated in terms of the three extremal lengths. To each \((R,\chi)\) and each modulus \(\tau\in M(R,\chi)\) a family \(\{(R_ t,\chi_ t)\}_{0\leq t<1}\) is obtained such that \((R,\chi)= (R_ 0,\chi_ 0)\) and \(M(R_ t,\chi_ t)\) shrinks continuously to \(\{\tau\}\) for \(t\to 1^ -\). Shiba's theorems concerning the area of \(T\backslash j(R)\) (\(j: R\to T\) a conformal embedding of \(R\) into a torus \(T\)) can thus be generalized.