Numerical Schottky uniformizations

Rubén A. Hidalgo, Jaime Figueroa

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3 Citations (Scopus)


A real algebraic curve of algebraic genus g ≥ 2 is a pair (S, τ), where S is a closed Riemann surface of genus g and τ is a reflection on S (anticonformal involution with fixed points). In this note, we discuss a numerical (Burnside) program which permits to obtain a Riemann period matrix of the surface S in terms of an uniformizing real Schottky group. If we denote by Aut+(S, τ) the group of conformal automorphisms of S commuting with the real structure τ, then it is a well known fact that |Aut +(S,τ)| ≤ 12(g-1). We say that (S,τ) is maximally symmetric if |Aut+(S,τ)|=12(g-1). We work explicitly such a numerical program in the case of maximally symmetric real curves of genus two. We construct a real Schottky uniformization for each such real curve and we use the numerical program to obtain a real algebraic curve, a Riemann period matrix and the accessory parameters in terms of the corresponding Schottky uniformization. In particular, we are able to give for Bolza's curve a Schottky uniformization (at least numerically), providing an example for which the inverse uniformization theorem is numerically solved. © Springer 2005.
Original languageEnglish
Pages (from-to)125-157
Number of pages33
JournalGeometriae Dedicata
Publication statusPublished - 1 Mar 2005

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