Convergence of Teichmüller deformations in the universal Teichmüller space

Abstract:

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