Reflections of regular maps and Riemann surfaces

Abstract

A compact Riemann surface of genus $g$ is called an M-surface if it admits an anti-conformal involution that fixes $g+1$ simple closed curves, the maximum number by Harnack's Theorem. Underlying every map on an orientable surface there is a Riemann surface and so the conclusions of Harnack's theorem still apply. Here we show that for each genus $gϯ1$ there is a unique M-surface of genus $g$ that underlies a regular map, and we prove a similar result for Riemann surfaces admitting anti-conformal involutions that fix $g$ curves.