Circular foliations and shear-radius coordinates on Teichm\"uller spaces of hyperbolic cone surfaces

Abstract

We study the Teichm\"uller space $\mathcal{T}(S,\underline{p})$ of hyperbolic cone-surfaces of fixed topological type with marked cone singularities. Fix a combinatorial triangulation $G$, and let $\mathcal{T}(G)\subset \mathcal{T}(S,\underline{p})$ be the locus where $G$ admits a geodesic realization; varying $G$, these loci form an open cover of $\mathcal{T}(S,\underline{p})$. On $\mathcal{T}(G)$ we construct a circular foliation adapted to geodesic triangular complementary regions, which is naturally decomposed into interior and peripheral parts. This decomposition defines shear parameters on edges and radius parameters at the singularities, and yields global coordinates on $\mathcal{T}(G)$: the resulting shear-radius map is a homeomorphism onto an explicit open cone in a finite-dimensional real vector space. In the spirit of Thurston, we then introduce partial stretch and anti-stretch deformations by rescaling the transverse measures of the interior or peripheral components. Peripheral stretch rays converge, in the simple-curve length-spectrum topology, to the cusped hyperbolic metric determined by the shear data, while interior anti-stretch rays converge to a circle-packed hyperbolic cone metric determined by the radii. Finally, we give criteria for the realization of prescribed cone angles for fixed $G$ and prove sharp upper bounds for admissible cone angles on the universally triangulable locus.