## On Harnack’s theorem and extensions: A geometric proof and applications

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- by Antonio F. Costa and Hugo Parlier PDF
- Conform. Geom. Dyn.
**12**(2008), 174-186 Request permission

## 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
- MR Author ID: 51935
- ORCID: 0000-0002-9905-0264
- Email: acosta@mat.uned.es
**Hugo Parlier**- Affiliation: Section de Mathématiques, École Polytechnique Fédérale de Lausanne, SB-IGAT, BCH, CH-1015 Lausanne, Switzerland
- MR Author ID: 767561
- Email: hugo.parlier@epfl.ch
- 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. - © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Conform. Geom. Dyn.
**12**(2008), 174-186 - MSC (2000): Primary 30F10, 32G15; Secondary 14H50, 30F20
- DOI: https://doi.org/10.1090/S1088-4173-08-00184-7
- MathSciNet review: 2448264