Open Access
November, 2008 Reflections of regular maps and Riemann surfaces
Adnan Melekoğlu , David Singerman
Rev. Mat. Iberoamericana 24(3): 921-939 (November, 2008).


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 $g ϯ 1$ 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.


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Adnan Melekoğlu . David Singerman . "Reflections of regular maps and Riemann surfaces." Rev. Mat. Iberoamericana 24 (3) 921 - 939, November, 2008.


Published: November, 2008
First available in Project Euclid: 9 December 2008

zbMATH: 1198.30041
MathSciNet: MR2490203

Primary: 05C10 , 30F10

Keywords: (M$-$1)-surface , M-surface , Platonic surface , regular map , Riemann surface

Rights: Copyright © 2008 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.24 • No. 3 • November, 2008
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