A compact Riemann surface of genus is called an M-surface if it admits an anti-conformal involution that fixes 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 there is a unique M-surface of genus that underlies a regular map, and we prove a similar result for Riemann surfaces admitting anti-conformal involutions that fix curves.
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Adnan Melekoğlu, David Singerman, Reflections of regular maps and Riemann surfaces. Rev. Mat. Iberoam. 24 (2008), no. 3, pp. 921–939DOI 10.4171/RMI/560