Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173

 
 

 

Bers embedding of the Teichmüller space of a once-punctured torus


Authors: Yohei Komori and Toshiyuki Sugawa
Journal: Conform. Geom. Dyn. 8 (2004), 115-142
MSC (2000): Primary 30F60; Secondary 30F40, 34A20
DOI: https://doi.org/10.1090/S1088-4173-04-00108-0
Published electronically: June 8, 2004
MathSciNet review: 2060380
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this note, we present a method of computing monodromies of projective structures on a once-punctured torus. This leads to an algorithm numerically visualizing the shape of the Bers embedding of a one-dimensional Teichmüller space. As a by-product, the value of the accessory parameter of a four-times punctured sphere will be calculated in a numerical way as well as the generators of a Fuchsian group uniformizing it. Finally, we observe the relation between the Schwarzian differential equation and Heun's differential equation in this special case.


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

  • 1. L. V. Ahlfors, Complex Analysis, 3rd ed., McGraw Hill, New York, 1979. MR 80c:30001
  • 2. A. F. Beardon, Iteration of Rational Functions, Springer-Verlag, 1991. MR 92j:30026
  • 3. L. Bers, Uniformization, moduli and Kleinian groups, Bull. London Math. Soc. 4 (1972), 257-300. MR 50:595
  • 4. A. Douady and C. J. Earle, Conformally natural extensions of homeomorphisms of the circle, Acta Math. 157 (1986), 23-48. MR 87j:30041
  • 5. D. B. A. Epstein and A. Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, Analytical and Geometric Aspects of Hyperbolic Space, London Math. Soc. Lecture Notes Series no. 111, Cambridge University Press, 1987, pp. 113-254. MR 89c:52014
  • 6. F. P. Gardiner, Schiffer's interior variation and quasiconformal mappings, Duke Math. 42 (1975), 371-380. MR 52:3519
  • 7. G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, fifth ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 81i:10002
  • 8. J. A. Hempel, The uniformization of the $n$-punctured sphere, Bull. London Math. Soc. 20 (1988), 97-115. MR 89c:30109
  • 9. E. Hille, Ordinary Differential Equations in the Complex Domain, Wiley-Interscience, 1976. MR 58:17266
  • 10. Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer-Tokyo, 1992. MR 94b:32031
  • 11. K. Ito, Exotic projective structures and quasi-Fuchsian space, Duke Math. J. 105 (2000), 185-209. MR 2001j:30038
  • 12. L. Keen, Teichmueller spaces of punctured tori: I, II, Complex Variables Theory Appl. 2 (1983), 199-211, and 213-225. MR 85b:32034
  • 13. L. Keen, H. E. Rauch, and A. T. Vasquez, Moduli of punctured tori and the accessory parameter of Lamé's equation, Trans. Amer. Math. Soc. 255 (1979), 201-230. MR 81j:30074
  • 14. L. Keen and C. Series, Pleating invariants for punctured torus groups, Warwick University preprint, 10/1998.
  • 15. -, Pleating coordinates for the Maskit embedding of the Teichmüller space of punctured tori, Topology 32 (1993), 719-749. MR 95g:32030
  • 16. S. P. Kerckhoff, Lines of minima in Teichmüller space, Duke Math. J. 65 (1992), 187-213. MR 93b:32027
  • 17. Y. Komori and C. Series, Pleating coordinates for the Earle embedding, Ann. Fac. Sci. Toulouse, VI. Sér., Math. 10 (2001), 69-105. MR 2004b:32021
  • 18. Y. Komori, T. Sugawa, M. Wada, and Y. Yamashita, Drawing Bers embeddings of the Teichmüller space of once-punctured tori, preprint.
  • 19. I. Kra, A generalization of a theorem of Poincaré, Proc. Amer. Math. Soc. 27 (1971), 299-302. MR 46:347
  • 20. -, Accessory parameters for punctured spheres, Trans. Amer. Math. Soc. 313 (1989), 589-617. MR 89j:30062
  • 21. I. Kra and B. Maskit, Remarks on projective structures, Riemann Surfaces and Related Topics, Ann. of Math. Studies, no. 97, 1981, pp. 343-359. MR 83f:30042
  • 22. S. L. Krushkal, Teichmüller spaces are not starlike, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), 167-173. MR 96f:32031
  • 23. I. Laine and T. Sorvali, Local solution of $w''+A(z)w=0$ and branched polymorphic functions, Result. Math. 10 (1986), 107-129. MR 87j:30109
  • 24. B. Maskit, On a class of Kleinian groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 442 (1969), 1-8. MR 40:5857
  • 25. C. McMullen, Rational maps and Kleinian groups, Proceedings of the International Congress of Mathematicians (Kyoto, 1990), Springer, 1991, pp. 889-900. MR 93h:57024
  • 26. -, Complex earthquakes and Teichmüller theory, J. Amer. Math. Soc. 11 (1998), 283-320. MR 98i:32030
  • 27. Y. Minsky, The classification of punctured-torus groups, Ann. of Math. (2) 149 (1999), 559-626. MR 2000f:30028
  • 28. H. Miyachi, Cusps in complex boundaries of one-dimensional Teichmüller spaces, Conform. Geom. Dyn. 7 (2003), 103-151.
  • 29. S. Nag, The Complex Analytic Theory of Teichmüller Spaces, Wiley, New York, 1988. MR 89f:32040
  • 30. Y. Okumura, Lifting problem of Fuchsian groups and a characterization of simple dividing loops on Riemann surfaces, in preparation.
  • 31. S. M. Pizer, Numerical Computing and Mathematical Analysis, Science Research Associates, 1975.
  • 32. R. M. Porter, Computation of a boundary point of Teichmüller space, Bol. Soc. Mat. Mexicana 24 (1979), 15-26. MR 81j:32026
  • 33. A. Ronveaux, Heun's Differential Equations, Oxford University Press, 1995. MR 98a:33005
  • 34. H. L. Royden, Automorphisms and isometries of Teichmüller space, Advances in the Theory of Riemann Surfaces, Ann. of Math. Stud. 66, Princeton Univ. Press, 1971, pp. 369-383. MR 44:5452
  • 35. H. Shiga, Projective structures on Riemann surfaces and Kleinian groups, J. Math. Kyoto Univ. 27 (1987), 433-438. MR 88k:30056
  • 36. H. Shiga and H. Tanigawa, Projective structures with discrete holonomy representations, Trans. Amer. Math. Soc. 351 (1999), 813-823. MR 99d:32025
  • 37. T. Sugawa, Estimates of hyperbolic metric with applications to Teichmüller spaces, Kyungpook Math. J. 42 (2002), 51-60. MR 2003g:30084
  • 38. M. Toki, On nonstarlikeness of Teichmüller spaces, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 58-60. MR 94h:32039
  • 39. P. Tukia, Quasiconformal extension of quasisymmetric mappings compatible with a Möbius group, Acta Math. 154 (1985), 153-193. MR 86f:30024
  • 40. D. J. Wright, The shape of the boundary of Maskit's embedding of the Teichmüller space of once punctured tori, unpublished manuscript.

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2000): 30F60, 30F40, 34A20

Retrieve articles in all journals with MSC (2000): 30F60, 30F40, 34A20


Additional Information

Yohei Komori
Affiliation: Department of Mathematics, Osaka City University, Sugimoto 3-3-138 Sumiyoshi-ku, Osaka, 558-8585 Japan
Email: komori@sci.osaka-cu.ac.jp

Toshiyuki Sugawa
Affiliation: Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526 Japan
Email: sugawa@math.sci.hiroshima-u.ac.jp

DOI: https://doi.org/10.1090/S1088-4173-04-00108-0
Keywords: Teichm\"uller space, Bers embedding, monodromy, pleating ray, accessory parameter, bending coordinates, once-punctured torus
Received by editor(s): November 6, 2003
Received by editor(s) in revised form: March 16, 2004
Published electronically: June 8, 2004
Additional Notes: The second author was partially supported by the Ministry of Education, Grant-in-Aid for Encouragement of Young Scientists, 9740056. A portion of the present research was carried out during the second author’s visit to the University of Helsinki under the exchange program of scientists between the Academy of Finland and the JSPS
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society