Convergence of Teichmüller deformations in the universal Teichmüller space

Abstract

Let $\phi :\mathbb{D}\to \mathbb{C}$ be an integrable holomorphic function on the unit disk $\mathbb{D}$ and $D_{\phi }:\mathbb{D}\to T(\mathbb{D})$ the Teichmüller disk in the universal Teichmüller space $T(\mathbb{D})$. For a positive $t$ it is known that $D_{\phi }(t)\to [{\mu }_{\phi }]\in PM{L}_b(\mathbb{D})$ as $t\to 1$, where ${\mu }_{\phi }$ is a bounded measured lamination representing a point on the Thurston boundary of $T(\mathbb{D})$. We extend this result by showing that $D_{\phi }:\mathbb{D}\to T(\mathbb{D})$ 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 $D_{\phi }(\lambda )$ when $\lambda \to e^{i\theta }$ is independent of the type of the approach to $e^{i\theta }\in \partial \mathbb{D}$.

14 pages, 1 figure