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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Areas of two-dimensional moduli spaces
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by Toshihiro Nakanishi and Marjatta Näätänen PDF
Proc. Amer. Math. Soc. 129 (2001), 3241-3252 Request permission


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
  • MR Author ID: 225488
  • Email:
  • Marjatta Näätänen
  • Affiliation: Department of Mathematics, University of Helsinki, P.O. Box 4 (Yliopistonkatu 5), 00014 Helsinki, Finland
  • Email:
  • 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
  • © Copyright 2001 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 129 (2001), 3241-3252
  • MSC (1991): Primary 32G15, 30F35, 57M50
  • DOI:
  • MathSciNet review: 1844999