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Convergence of Teichmüller deformations in the universal Teichmüller space


Authors: Hideki Miyachi and Dragomir Šarić
Journal: Proc. Amer. Math. Soc. 147 (2019), 4877-4889
MSC (2010): Primary 30F60; Secondary 30C62, 30L99
DOI: https://doi.org/10.1090/proc/14598
Published electronically: May 17, 2019
MathSciNet review: 4011520
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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}$.


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Additional Information

Hideki Miyachi
Affiliation: Division of Mathematical and Physical Sciences, Graduate School of Natural Science & Technology, Kanazawa University, Kakuma-machi, Kanazawa, Ishikawa, 920-1192, Japan
Email: miyachi@se.kanazawa-u.ac.jp

Dragomir Šarić
Affiliation: Department of Mathematics, Queens College of CUNY, 65-30 Kissena Boulevard, Flushing, New York 11367; and Mathematics Ph.D. Program, The CUNY Graduate Center, 365 Fifth Avenue, New York, New York 10016-4309
Email: Dragomir.Saric@qc.cuny.edu

DOI: https://doi.org/10.1090/proc/14598
Received by editor(s): October 3, 2018
Received by editor(s) in revised form: February 19, 2019
Published electronically: May 17, 2019
Additional Notes: The first author was partially supported by JSPS KAKENHI Grant Numbers 16K05202, 16H03933, 17H02843.
The second author was partially supported by a Simons Foundation grant, Grant Number 346391.
Communicated by: Ken Bromberg
Article copyright: © Copyright 2019 American Mathematical Society