The Weil-Petersson K\"ahler form and affine foliations on surfaces

TL;DR

This paper extends classical geometric structures like the Weil-Petersson form to broken hyperbolic structures on surfaces, establishing their boundary relations and limits using techniques from decorated Teichmüller theory.

Contribution

It introduces broken analogues of the Weil-Petersson form and Thurston symplectic form and proves their boundary relationship, expanding the understanding of hyperbolic surface structures.

Findings

Extension of Weil-Petersson form to broken hyperbolic structures

Boundary relationship between broken structures and measured foliations

Application of decorated Teichmüller theory techniques

Abstract

The space of broken hyperbolic structures generalizes the Teichm\"uller space of a punctured surface, and the space of projectivized broken measured foliations (equivalently, the space of projectivized affine foliations) generalizes the space of projectivized measured foliations. Just as projectivized measured foliations provide Thurston's boundary for Teichm\"uller space, so too do projectivized broken measured foliations form a boundary for the space of broken hyperbolic structures. In this paper, we naturally extend the Weil-Petersson K\"ahler two-form and the Thurston symplectic form to their broken analogues and prove that the former suitably limits to the latter. The proof in sketch follows earlier work of the authors for measured foliations and depends upon techniques from decorated Teichm\"uller theory, which is also applied here to a further study of broken hyperbolic structures.