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Conformal Geometry and Dynamics

ISSN 1088-4173



On Harnack's theorem and extensions: A geometric proof and applications

Authors: Antonio F. Costa and Hugo Parlier
Journal: Conform. Geom. Dyn. 12 (2008), 174-186
MSC (2000): Primary 30F10, 32G15; Secondary 14H50, 30F20
Published electronically: October 16, 2008
MathSciNet review: 2448264
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Abstract: Harnack's theorem states that the fixed points of an orientation reversing involution of a compact orientable surface of genus $ g$ are a set of $ k$ disjoint simple closed geodesic where $ 0\leq k\leq g+1$. The first goal of this article is to give a purely geometric, complete and self-contained proof of this fact. In the case where the fixed curves of the involution do not separate the surface, we prove an extension of this theorem, by exhibiting the existence of auxiliary invariant curves with interesting properties. Although this type of extension is well known (see, for instance, Comment. Math. Helv. 57(4): 603-626 (1982) and Transl. Math. Monogr., vol. 225, Amer. Math. Soc., Providence, RI, 2004), our method also extends the theorem in the case where the surface has boundary. As a byproduct, we obtain a geometric method on how to obtain these auxiliary curves. As a consequence of these constructions, we obtain results concerning presentations of Non-Euclidean crystallographic groups and a new proof of a result on the set of points corresponding to real algebraic curves in the compactification of the Moduli space of complex curves of genus $ g$, $ \overline{\mathcal{M}_{g}}$. More concretely, we establish that given two real curves there is a path in $ \overline{\mathcal{M}_{g}}$ which passes through at most two singular curves, a result of M. Seppälä (Ann. Sci. École Norm. Sup. (4), 24(5), 519-544 (1991)).

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Additional Information

Antonio F. Costa
Affiliation: Departamento de Matemáticas Fundamentales, Facultad de Ciencias, Universidad Nacional de Educación a Distancia, Madrid 28040, Spain

Hugo Parlier
Affiliation: Section de Mathématiques, École Polytechnique Fédérale de Lausanne, SB-IGAT, BCH, CH-1015 Lausanne, Switzerland

Keywords: Orientation reversing involutions, simple closed geodesics, hyperbolic Riemann surfaces
Received by editor(s): June 25, 2007
Published electronically: October 16, 2008
Additional Notes: The first author was supported in part by BFM 2002-04801
The second author was supported in part by SNFS grant number PBEL2-106180.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.