Remote Access Conformal Geometry and Dynamics
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. 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, 10.1007/BF02567818
  • 2. Peter Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, vol. 106, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1183224
  • 3. 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
  • 4. 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) Princeton Univ. Press, Princeton, N.J., 1971, pp. 119–130. Ann. of Math. Studies, No. 66. MR 0296282
  • 5. Axel Harnack, Ueber die Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann. 10 (1876), no. 2, 189–198 (German). MR 1509883, 10.1007/BF01442458
  • 6. Felix Klein, Eine neue Relation zwischen den Singularitäten einer algebraischen Curve, Math. Ann. 10 (1876), no. 2, 199–209 (German). MR 1509884, 10.1007/BF01442459
  • 7. Irwin Kra and Bernard Maskit, Bases for quadratic differentials, Comment. Math. Helv. 57 (1982), no. 4, 603–626. MR 694607, 10.1007/BF02565877
  • 8. A. M. Macbeath, The classification of non-euclidean plane crystallographic groups, Canad. J. Math. 19 (1967), 1192–1205. MR 0220838
  • 9. 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.
  • 10. P. Schmutz, Riemann surfaces with shortest geodesic of maximal length, Geom. Funct. Anal. 3 (1993), no. 6, 564–631. MR 1250756, 10.1007/BF01896258
  • 11. M. Seppälä, Moduli spaces of stable real algebraic curves, Ann. Sci. École Norm. Sup. (4) 24 (1991), no. 5, 519–544. MR 1132756
  • 12. 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.
  • 13. H. C. Wilkie, On non-Euclidean crystallographic groups, Math. Z. 91 (1966), 87–102. MR 0185013

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2000): 30F10, 32G15, 14H50, 30F20

Retrieve articles in all journals with MSC (2000): 30F10, 32G15, 14H50, 30F20

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.