Convergence of Teichm\"uller deformations in the universal Teichm\"uller space

TL;DR

This paper extends the understanding of Teichmüller disk convergence in the universal Teichmüller space, showing continuous extension to the boundary and independence of convergence rate on approach direction.

Contribution

It proves the continuous extension of Teichmüller disks to the Thurston boundary and establishes the uniformity of convergence rate regardless of boundary approach.

Findings

Teichmüller disks extend continuously to the Thurston boundary.

Convergence rate of Teichmüller disks is independent of approach direction.

The results generalize known boundary convergence properties in Teichmüller theory.

Abstract

Let $\varphi :\mathbb{D}\to\mathbb{C}$ be an integrable holomorphic function on the unit disk $\mathbb{D}$ and $D_{\varphi}:\mathbb{D}\to T(\mathbb{D})$ the Teichm\"uller disk in the universal Teichm\"uller 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 $e^{i\theta}\in\partial\mathbb{D}$.