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

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- by Antonio F. Costa and Hugo Parlier
- Conform. Geom. Dyn.
**12**(2008), 174-186 - DOI: https://doi.org/10.1090/S1088-4173-08-00184-7
- Published electronically: October 16, 2008
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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)).

## References

- P. Buser, M. Seppälä, and R. Silhol,
*Triangulations and moduli spaces of Riemann surfaces with group actions*, Manuscripta Math.**88**(1995), no. 2, 209–224. MR**1354107**, DOI 10.1007/BF02567818 - Peter Buser,
*Geometry and spectra of compact Riemann surfaces*, Progress in Mathematics, vol. 106, Birkhäuser Boston, Inc., Boston, MA, 1992. MR**1183224** - A. F. Costa and M. Izquierdo,
*On the locus of real algebraic curves*, Atti Sem. Mat. Fis. Univ. Modena**49**(2001), no. suppl., 91–107. Dedicated to the memory of Professor M. Pezzana (Italian). MR**1881092** - Clifford J. Earle,
*On the moduli of closed Riemann surfaces with symmetries*, Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969) Ann. of Math. Studies, No. 66, Princeton Univ. Press, Princeton, N.J., 1971, pp. 119–130. MR**0296282** - Axel Harnack,
*Ueber die Vieltheiligkeit der ebenen algebraischen Curven*, Math. Ann.**10**(1876), no. 2, 189–198 (German). MR**1509883**, DOI 10.1007/BF01442458 - Felix Klein,
*Eine neue Relation zwischen den Singularitäten einer algebraischen Curve*, Math. Ann.**10**(1876), no. 2, 199–209 (German). MR**1509884**, DOI 10.1007/BF01442459 - Irwin Kra and Bernard Maskit,
*Bases for quadratic differentials*, Comment. Math. Helv.**57**(1982), no. 4, 603–626. MR**694607**, DOI 10.1007/BF02565877 - A. M. Macbeath,
*The classification of non-euclidean plane crystallographic groups*, Canadian J. Math.**19**(1967), 1192–1205. MR**220838**, DOI 10.4153/CJM-1967-108-5 - S. M. Natanzon.
*Moduli of Riemann surfaces, real algebraic curves, and their superanalogs*, volume 225 of*Translations of Mathematical Monographs*. American Mathematical Society, Providence, RI, 2004. Translated from the 2003 Russian edition by Sergei Lando. - P. Schmutz,
*Riemann surfaces with shortest geodesic of maximal length*, Geom. Funct. Anal.**3**(1993), no. 6, 564–631. MR**1250756**, DOI 10.1007/BF01896258 - M. Seppälä,
*Moduli spaces of stable real algebraic curves*, Ann. Sci. École Norm. Sup. (4)**24**(1991), no. 5, 519–544. MR**1132756**, DOI 10.24033/asens.1635 - G. Weichold.
*Ueber symmetrische Riemann’sche Flächen und die Periodicitätsmoduln der zugehörigen Abel’schen Normalintegrale erster Gattung.*PhD thesis, Diss. Leipzig. Schlömilch Z. XXVIII. 321-352, 1883. - H. C. Wilkie,
*On non-Euclidean crystallographic groups*, Math. Z.**91**(1966), 87–102. MR**185013**, DOI 10.1007/BF01110157

## Bibliographic 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