On the inverse uniformization problem: real Schottky uniformizations
There are several ways to describe closed Riemann surfaces, for example, Fuchsian uniformizations, Schottky uniformizations, algebraic curves, Riemann period matrices. In this paper the author studies the inverse uniformization problem, i.e. a problem to provide a Fuchsian group, a Schottky group, an algebraic curve and a Riemann period matrix corresponding to the same Riemann surface. So far the problem has been considered explicitly or numerically, and several results are obtained for mostly hyperelliptic surfaces. The author considers this problem for a family \(\mathcal{S}_{n,k}\) of pairs \((S,H)\), where \(S\) is a real Riemann surface constructed by \(2k\) copies of a hyperbolic \(n\)-gon with an angle \(\pi/k\) and the others \(\pi/2\); \(H\) is an automorphism group of \(S\) isomorphic to \(\mathbb{Z}_2\oplus \mathcal{D}_k\) (\(\mathcal{D}_k\) denotes the dihedral group of order \(2k\)). In general, the surfaces are non-hyperelliptic. First, the pairs \((S,H)\) are described in terms of Fuchsian uniformization. Second, it is shown that \((S,H)\) can be described in terms of Schottky uniformization by introducing circular polygons. Third, the author shows a way to construct a Riemann period matrix from the Schottky uniformization derived from a circular polygon, and gives examples to construct Riemann period matrices and algebraic curves when \((n,k)=(5,2)\) (genus two surface) and \((n,k)=(5,3)\) (genus three surface).
scientific article; zbMATH DE number 5968180
scientific article