The Weil-Petersson symplectic structure at Thurston’s boundary

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.