Reflections of regular maps and Riemann surfaces

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 g> 1 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.