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On the inverse uniformization problem: real Schottky uniformizations

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Abstract

Classical uniformization theorem, together with Torelli’s theorem, assert that each closed Riemann surface may be described in terms of Fuchsian uniformizations (highests uniformizations), Schottky uniformizations (lowest uniformizations), algebraic curves, Riemann period matrices, etc. The inverse uniformization problem may be stated as to provide any or some of the other descriptions once one of them is given. In general (part of) this inverse problem has been (numerically) partially solved for hyperelliptic Riemann surfaces.

In this paper we provide a family of real Riemann surfaces, in general non-hyperelliptic ones, which are described in terms of Fuchsian and Schottky uniformizations. The explicit relation between these two uniformizations is given. The Schottky uniformization is used to compute a suitable Riemann period matrix of the uniformized surface so that its coefficients are given in terms of the corresponding Schottky group. Two explicit examples, one of genus 2 and the other of genus 3 (non-hyperelliptic), are provided and (numerically) algebraic curve representations are given for them.

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  1. Departamento de Matemáticas, Universidad Técnica Federico Santa María, Valparaíso, Chile

    Rubén A. Hidalgo

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  1. Rubén A. Hidalgo
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Correspondence to Rubén A. Hidalgo.

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Research partially supported by projects Fondecyt 1070271 and UTFSM 12.09.02.

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Hidalgo, R.A. On the inverse uniformization problem: real Schottky uniformizations. Rev Mat Complut 24, 391–420 (2011). https://doi.org/10.1007/s13163-010-0046-3

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