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Areas of two-dimensional moduli spaces

Authors: Toshihiro Nakanishi and Marjatta Näätänen
Journal: Proc. Amer. Math. Soc. 129 (2001), 3241-3252
MSC (1991): Primary 32G15, 30F35, 57M50
Published electronically: April 2, 2001
MathSciNet review: 1844999
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Abstract: Wolpert's formula expresses the Weil-Petersson $2$-form in terms of the Fenchel-Nielsen coordinates in case of a closed or punctured surface. The area-form in Fenchel-Nielsen coordinates is invariant under the mapping class group on each 2-dimensional Teichmüller space of a surface with singularities, hence areas with respect to it can be calculated for 2-dimensional moduli spaces in cases when the Teichmüller space admits global Fenchel-Nielsen coordinates: The area of the moduli space for the signature $(0;2\theta _{1},2\theta _{2},2\theta _{3},2\theta _{4})$ is $2(\pi ^{2}-\theta _{1}^{2}-\theta _{2}^{2}-\theta _{3}^{2}-\theta _{4}^{2})$, the definition of signature is generalized to include punctures, cone points and geodesic boundary curves. In case the surface is represented by a Fuchsian group, the area is the classical Weil-Petersson area.

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Additional Information

Toshihiro Nakanishi
Affiliation: Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-01, Japan

Marjatta Näätänen
Affiliation: Department of Mathematics, University of Helsinki, P.O. Box 4 (Yliopistonkatu 5), 00014 Helsinki, Finland

Received by editor(s): September 23, 1999
Received by editor(s) in revised form: March 9, 2000
Published electronically: April 2, 2001
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2001 American Mathematical Society