Abstract

Let S be a real closed Riemann surfaces together a reflection ?: S ? S, that is, an anticonformal involution with fixed points. A well known fact due to C.L. May [19] asserts that the group K(S, ?), consisting on all automorphisms (conformal and anticonformal) of S which commutes with ?, has order at most 24(g - 1). The surface S is called maximally symmetric Riemann surface if ?K(S, ?) = 24(g - 1) [8]. In this note we proceed to construct real Schottky uniformizations of all maximally symmetric Riemann surfaces of genus g ? 5. A method due to Burnside [3] permits us the computation of a basis of holomorphic one forms in terms of these real Schottky groups and, in particular, to compute a Riemann period matrix for them. We also use this in genus 2 and 3 to compute an algebraic curve representing the uniformized surface S. The arguments used in this note can be programed into a computer program in order to obtain numerical approximation of Riemann period matrices and algebraic curves for the uniformized surface S in terms of the parameters defining the real Schottky groups.