Abstract
The universal cover or the covering group of a hyperbolic Riemann surface \(X\) is important but hard to express explicitly. It can be, however, detected by the uniformization and a suitable description of \(X\). Beardon proposed five different ways to describe twice-punctured disks using fundamental domain, hyperbolic length, collar and extremal length in 2012. We parameterize a once-punctured annulus \(A\) in terms of five parameter pairs and give explicit formulas about the hyperbolic structure and the complex structure of \(A\). Several degenerate cases are also treated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.: Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley, New York (1997)
Basmajian, A.: Constructing pairs of pants. Ann. Acad. Sci. Fenn. Ser. A. I. Math. 15, 65–74 (1990)
Beardon, A.F.: Graduate texts in mathematics. In: The Geometry of Discrete Groups, vol. 91. Springer, Berlin (1983)
Beardon, A.F.: The uniformisation of a twice-punctured disc. Comput. Methods Funct. Theory 12(2), 585–596 (2012)
Buser, P.: Geometry and spectra of compact Riemann surfaces. In: Progress in Mathematics, vol. 106. Birkhäuser Boston Inc., Boston (1992)
Byrd, P.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Physicists. Springer, Berlin (1954)
Farkas, H.M., Kra, I.: Graduate studies in mathematics. In: Theta Constants, Riemann Surfaces and the Modular Group, vol. 37. American Mathematical Society, Providence (2001)
Hejhal, D.A.: Universal covering maps for variable regions. Math. Z. 137, 7–20 (1974)
Hempel, J.A., Smith, S.J.: Hyperbolic lengths of geodesics surrounding two punctures. Proc. Am. Math. Soc. 103, 513–516 (1988)
Hempel, J.A., Smith, S.J.: The accessory parameter problem for the uniformization of the twice-punctured disc. J. Lond. Math. Soc. 40(2), 269–279 (1989)
Hempel, J.A., Smith, S.J.: Uniformisation of the twice-punctured disc-problems of confluence. Bull. Aust. Math. Soc. 39, 369–387 (1989)
Keen, L., Lakic, N.: Hyperbolic geometry from a local viewpoint. In: London Mathematical Society Student Texts, vol. 68. Cambridge University Press, Cambridge (2007)
Lehto, O., Virtanen, K.I.: Quasiconformal mappings in the plane. Grundlehren Math., Wiss., Band, 2nd edn. Springer, New York (1973)
Nevanlinna, R.: Analytic Functions. Springer, Berlin (1970)
Maskit, B.: Comparison of hyperbolic and extremal lengths. Ann. Acad. Sci. Fenn. Ser. A. I. Math. 10, 381–386 (1985)
Ohtsuka, M.: Dirichlet Problems, Extremal Length and Prime ends. Van Nostrand, New York (1970)
Sugawa, T.: Various domains constants related to uniform perfectness. Complex Var. 36, 311–345 (1998)
Acknowledgments
The author would like to thank Prof. Toshiyuki Sugawa for his proposal for this topic, his suggestions and encouragements. The author is grateful to Prof. Katsuhiko Matsuzaki for his idea in Remark 3, and Prof. Ara Basmajian for his comments on the hyperbolic and complex structures of \(A\). The author also thanks the referee for his/her valuable suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Mario Bonk.
Rights and permissions
About this article
Cite this article
Zhang, T. Uniformization of a Once-Punctured Annulus. Comput. Methods Funct. Theory 15, 75–91 (2015). https://doi.org/10.1007/s40315-014-0095-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40315-014-0095-6