No CrossRef data available.
Article contents
Some results on Teichmüller spaces of Klein surfaces
Published online by Cambridge University Press: 18 May 2009
Extract
Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
The deformation theory of nonorientable surfaces deals with the problem of studying parameter spaces for the different dianalytic structures that a surface can have. It is an extension of the classical theory of Teichmüller spaces of Riemann surfaces, and as such, it is quite rich. In this paper we study some basic properties of the Teichmüller spaces of non-orientable surfaces, whose parallels in the orientable situation are well known. More precisely, we prove an uniformization theorem, similar to the case of Riemann surfaces, which shows that a non-orientable compact surface can be represented as the quotient of a simply connected domain of the Riemann sphere, by a discrete group of Möbius and anti-Möbius transformation (mappings whose conjugates are Mobius transformations). This uniformization result allows us to give explicit examples of Teichmüller spaces of non-orientable surfaces, as subsets of deformation spaces of orientable surfaces. We also prove two isomorphism theorems: in the first place, we show that the Teichmüller spaces of surfaces of different topological type are not, in general, equivalent. We then show that, if the topological type is preserved, but the signature changes, then the deformations spaces are isomorphic. These are generalizations of the Patterson and Bers-Greenberg theorems for Teichmüller spaces of Riemann surfaces, respectively.
- Type
- Research Article
- Information
- Copyright
- Copyright © Glasgow Mathematical Journal Trust 1997
References
REFERENCES
1.Abikoff, W., The real analytic theory of Teichmüller space Lecture Notes in Mathematics, No 820, (Springer-Verlag, 1980).Google Scholar
2.Alling, N. and Greenleaf, N., Foundations of the theory of Klein surfaces, Lectures Notes in Mathematics, No 219, (Springer-Verlag, 1971).Google Scholar
3.Arés-Gastesi, P., The Bers-Greenberg theorem and the Maskit embedding for Teichmüller spaces, To appear in Bull. London Math. Soc.Google Scholar
4.Bers, L., Nielsen extensions of Riemann surfaces, Annal Acad. Sci. Fenn. Ser. A l Math. 2 ((1976), 197–202.Google Scholar
5.Bujalance, E., Costa, A. F., Natanzon, S. M., and Singerman, D., Involutions of Klein Surfaces, Math. Z. 211 (1992), 461–478.CrossRefGoogle Scholar
6.Bujalance, E., Etayo, J. J. and Gamboa, J. M., Hyperelliptic Klein Surfaces, Quart. J. Math. Oxford Ser. 2 36 (1985), 141–157.CrossRefGoogle Scholar
7.Bujalance, E. and Singerman, D., The symmetry type of a Riemann surface, Proc. London Math. Soc. 51 (1985), 501–519.Google Scholar
8.Earle, C. and Kra, I., On holomorphic mappings between Teichmüller spaes, Contributions to Analysis (Academic Press, 1974), 107–124.Google Scholar
9.Greenleaf, N. and Read, W., Positive holomorphic differentials on Klein surfaces, Pacific J. Math. 32 (1970), 711–713.Google Scholar
11.Kra, I., Non-variational global coordinates for Teichmüller spaces, Holomorphic functions and Moduli II, Math. Sci. Res. Inst. Pubi., 11, (Springer, 1988), pp. 221–249.Google Scholar
12.Kra, I., Horocyclic coordinates for Riemann surfaces and moduli spaces. I; Teichmüller and Riemann spaces of Kleinian groups, J. Amer. Math. Soc. 3 (1990), 499–578.Google Scholar
13.Maskit, B., On boundaries of Teichmuller spaces and on Kleinian groups: II, Ann. of Math. (2) 91 (1970), 607–639.Google Scholar
14.Maskit, B., On the classification of Kleinian groups: I-Koebe groups, Acta Math. 135 (1975), 249–270.CrossRefGoogle Scholar
16.Seppälä, M., Teichmüller Spaces of Klein Surfaces, Ann. Acad. Sci. Fenn. Ser. Al Malhematica Dissertationes 15(1978), 1–37.Google Scholar
17.Seppälä, M. and Sorvali, T., Geometry of Riemann Surfaces and Teichmüller Spaces, (North-Holland, 1992).Google Scholar
You have
Access