Convergence of Teichmüller deformations in the universal Teichmüller space

Abstract

Let $φ:\mathbb{D}\to\mathbb{C}$ be an integrable holomorphic function on the unit disk $\mathbb{D}$ and $D_φ:\mathbb{D}\to T(\mathbb{D})$ the Teichmüller disk in the universal Teichmüller space $T(\mathbb{D})$. For a positive $t$ it is known that $D_φ(t)\to [μ_φ]\in PML_b(\mathbb{D})$ as $t\to 1$, where $μ_φ$ is a bounded measured lamination representing a point on the Thurston boundary of $T(\mathbb{D})$. We extend this result by showing that $D_φ\colon \mathbb{D}\to T(\mathbb{D})$ extends as a continuous map from the closed disk $\overline{\mathbb{D}}$ to the Thurston bordification. In addition, we prove that the rate of convergence of $D_φ(λ)$ when $λ\to e^{iθ}$ is independent of the type of the approach to $e^{iθ}\in\partial\mathbb{D}$.