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Conformal Geometry and Dynamics

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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
Published electronically: June 8, 2004
MathSciNet review: 2060380
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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.

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Additional Information

Yohei Komori
Affiliation: Department of Mathematics, Osaka City University, Sugimoto 3-3-138 Sumiyoshi-ku, Osaka, 558-8585 Japan

Toshiyuki Sugawa
Affiliation: Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526 Japan

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

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