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

Komori, Yohei; Sugawa, Toshiyuki

doi:10.1090/S1088-4173-04-00108-0

Bers embedding of the Teichmüller space of a once-punctured torus
HTML articles powered by AMS MathViewer

by Yohei Komori and Toshiyuki Sugawa

Conform. Geom. Dyn. 8 (2004), 115-142

DOI: https://doi.org/10.1090/S1088-4173-04-00108-0

Published electronically: June 8, 2004

PDF | Request permission

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

Lars V. Ahlfors, Complex analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable. MR 510197
Alan F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. MR 1128089, DOI 10.1007/978-1-4612-4422-6
Lipman Bers, Uniformization, moduli, and Kleinian groups, Bull. London Math. Soc. 4 (1972), 257–300. MR 348097, DOI 10.1112/blms/4.3.257
Adrien Douady and Clifford J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), no. 1-2, 23–48. MR 857678, DOI 10.1007/BF02392590
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 (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 113–253. MR 903852
Frederick P. Gardiner, Schiffer’s interior variation and quasiconformal mapping, Duke Math. J. 42 (1975), 371–380. MR 382637
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
Joachim A. Hempel, On the uniformization of the $n$-punctured sphere, Bull. London Math. Soc. 20 (1988), no. 2, 97–115. MR 924235, DOI 10.1112/blms/20.2.97
Einar Hille, Ordinary differential equations in the complex domain, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1976. MR 0499382
Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors. MR 1215481, DOI 10.1007/978-4-431-68174-8
Kentaro Ito, Exotic projective structures and quasi-Fuchsian space, Duke Math. J. 105 (2000), no. 2, 185–209. MR 1793610, DOI 10.1215/S0012-7094-00-10521-2
Linda Keen, Teichmueller spaces of punctured tori. I, II, Complex Variables Theory Appl. 2 (1983), no. 2, 199–211, 213–225. MR 725269, DOI 10.1080/17476938308814042
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 542877, DOI 10.1090/S0002-9947-1979-0542877-9

Pleating invariants for punctured torus groups

Linda Keen and Caroline Series, Pleating coordinates for the Maskit embedding of the Teichmüller space of punctured tori, Topology 32 (1993), no. 4, 719–749. MR 1241870, DOI 10.1016/0040-9383(93)90048-Z
Steven P. Kerckhoff, Lines of minima in Teichmüller space, Duke Math. J. 65 (1992), no. 2, 187–213. MR 1150583, DOI 10.1215/S0012-7094-92-06507-0
Yohei Komori and Caroline Series, Pleating coordinates for the Earle embedding, Ann. Fac. Sci. Toulouse Math. (6) 10 (2001), no. 1, 69–105 (English, with English and French summaries). MR 1928990, DOI 10.5802/afst.985

Drawing Bers embeddings of the Teichmüller space of once-punctured tori

Irwin Kra, A generalization of a theorem of Poincaré, Proc. Amer. Math. Soc. 27 (1971), 299–302. MR 301189, DOI 10.1090/S0002-9939-1971-0301189-7
Irwin Kra, Accessory parameters for punctured spheres, Trans. Amer. Math. Soc. 313 (1989), no. 2, 589–617. MR 958896, DOI 10.1090/S0002-9947-1989-0958896-0
Irwin Kra and Bernard Maskit, Remarks on projective structures, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 343–359. MR 624824
Samuel L. Krushkal, Teichmüller spaces are not starlike, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), no. 1, 167–173. MR 1304114
Ilpo Laine and Tuomas Sorvali, Local solutions of $w''+A(z)w=0$ and branched polymorphic functions, Results Math. 10 (1986), no. 1-2, 107–129. MR 869803, DOI 10.1007/BF03322368
Bernard Maskit, On a class of Kleinian groups, Ann. Acad. Sci. Fenn. Ser. A I 442 (1969), 8. MR 0252638
Curt McMullen, Rational maps and Kleinian groups, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 889–899. MR 1159274
Curtis T. McMullen, Complex earthquakes and Teichmüller theory, J. Amer. Math. Soc. 11 (1998), no. 2, 283–320. MR 1478844, DOI 10.1090/S0894-0347-98-00259-8
Yair N. Minsky, The classification of punctured-torus groups, Ann. of Math. (2) 149 (1999), no. 2, 559–626. MR 1689341, DOI 10.2307/120976

Cusps in complex boundaries of one-dimensional Teichmüller spaces

7

Subhashis Nag, The complex analytic theory of Teichmüller spaces, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1988. A Wiley-Interscience Publication. MR 927291

Lifting problem of Fuchsian groups and a characterization of simple dividing loops on Riemann surfaces

Numerical Computing and Mathematical Analysis

R. Michael Porter, Computation of a boundary point of Teichmüller space, Bol. Soc. Mat. Mexicana (2) 24 (1979), no. 1, 15–26. MR 579666
A. Ronveaux (ed.), Heun’s differential equations, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by F. M. Arscott, S. Yu. Slavyanov, D. Schmidt, G. Wolf, P. Maroni and A. Duval. MR 1392976
H. L. Royden, Automorphisms and isometries of Teichmüller space, 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. 369–383. MR 0288254
Hiroshige Shiga, Projective structures on Riemann surfaces and Kleinian groups, J. Math. Kyoto Univ. 27 (1987), no. 3, 433–438. MR 910228, DOI 10.1215/kjm/1250520657
Hiroshige Shiga and Harumi Tanigawa, Projective structures with discrete holonomy representations, Trans. Amer. Math. Soc. 351 (1999), no. 2, 813–823. MR 1443890, DOI 10.1090/S0002-9947-99-02043-7
Toshiyuki Sugawa, Estimates of hyperbolic metric with applications to Teichmüller spaces, Kyungpook Math. J. 42 (2002), no. 1, 51–60. MR 1915906
Masahiko Toki, On nonstarlikeness of Teichmüller spaces, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 3, 58–60. MR 1222823
Pekka Tukia, Quasiconformal extension of quasisymmetric mappings compatible with a Möbius group, Acta Math. 154 (1985), no. 3-4, 153–193. MR 781586, DOI 10.1007/BF02392471

The shape of the boundary of Maskit’s embedding of the Teichmüller space of once punctured tori