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Transactions of the American Mathematical Society

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The Weil-Petersson symplectic structure at Thurston's boundary

Authors: A. Papadopoulos and R. C. Penner
Journal: Trans. Amer. Math. Soc. 335 (1993), 891-904
MSC: Primary 57M50; Secondary 30F60, 32G15
MathSciNet review: 1089420
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Abstract: The Weil-Petersson Kähler structure on the Teichmüller space $ \mathcal{T}$ of a punctured surface is shown to extend, in an appropriate sense, to Thurston's symplectic structure on the space $ \mathcal{M}{\mathcal{F}_0}$ of measured foliations of compact support on the surface. We introduce a space $ {\widetilde{\mathcal{M}\mathcal{F}}_0}$ of decorated measured foliations whose relationship to $ \mathcal{M}{\mathcal{F}_0}$ is analogous to the relationship between the decorated Teichmüller space $ \tilde{\mathcal{T}}$ and $ \mathcal{T}$. $ \widetilde{\mathcal{M}{\mathcal{F}_0}}$ is parametrized by a vector space, and there is a natural piecewise-linear embedding of $ \mathcal{M}{\mathcal{F}_0}$ in $ \widetilde{\mathcal{M}{\mathcal{F}_0}}$ which pulls back a global differential form to Thurston's symplectic form. We exhibit a homeomorphism between $ \tilde{\mathcal{T}}$ and $ {\widetilde{\mathcal{M}\mathcal{F}}_0}$ which preserves the natural two-forms on these spaces. Following Thurston, we finally consider the space $ \mathcal{Y}$ of all suitable classes of metrics of constant Gaussian curvature on the surface, form a natural completion $ \overline{\mathcal{Y}}$ of $ \mathcal{Y}$, and identify $ \overline{\mathcal{Y}} - \mathcal{Y}$ with $ \mathcal{M}{\mathcal{F}_0}$. An extension of the Weil-Petersson Kähler form to $ \mathcal{Y}$ is found to extend continuously by Thurston's symplectic pairing on $ \mathcal{M}{\mathcal{F}_0}$ to a two-form on $ \overline{\mathcal{Y}}$ itself.

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