The Weil-petersson Kähler Form and Affine Foliations on Surfaces

The space of broken hyperbolic structures generalizes the usual Teichmüller space of a punctured surface, and the space of projectivized broken measured foliations–or equivalently, the space of projectivized affine foliations of a punctured surface–likewise admits a generalization to projectivized broken measured foliations. Just as projectivized measured foliations provide Thurston's boundary for Teichmüller space, so too do projectivized broken measured foliations provide a boundary for the space of broken hyperbolic structures. In this paper, we naturally extend the Weil-Petersson Kähler two-form to a corresponding two-form on the space of broken hyperbolic structures as well as Thurston's symplectic form to a corresponding two-form on the space of broken measured foliations, and we show that the former limits in an appropriate sense to the latter. The proof in sketch follows earlier work of the authors for measured foliations and depends upon techniques from decorated Teichmüller theory, which is also applied here to a further study of broken hyperbolic structures. 0.— Introduction In the paper [PP], we established a relation between the Weil-Petersson Kähler form on the Teichmüller space of a punctured surface and Thurston's piecewise-linear symplectic form on the space of measured foliations of compact support on that surface. We extend here this work to the context of broken hyperbolic structures and of broken measured foliations. Both spaces of broken structures were defined in [OP2], and we shall recall the definitions in the next section. The space of broken hyperbolic structures contains Teichmüller space as a proper subspace; the space of broken measured foliations can be identified with the space of affine foliations on the surface (as discussed in Appendix A), and it likewise contains the space of measured foliations as a proper subspace. More precisely, in this paper, we define a two-form on the space of broken hyperbolic structures which extends the Weil-Petersson Kähler two-form defined on Teichmüller space. Likewise, we define a two-form on the space of broken measured foliations, which extends Thurston's form defined on measured foliations space. In 1