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Asymptotic Teichmüller space of a closed set of the Riemann sphere


Authors: Yi Qi and Yan Wu
Journal: Proc. Amer. Math. Soc. 146 (2018), 2867-2876
MSC (2010): Primary 32G15; Secondary 30C62, 30F60
DOI: https://doi.org/10.1090/proc/13144
Published electronically: March 14, 2018
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Abstract: The asymptotic Teichmüller space $ AT(E)$ of a closed subset $ E$ of the Riemann sphere $ \hat {\mathbb{C}}$ with at least $ 4$ points and the natural asymptotic Teichmüller metric are introduced. It is proved that $ AT(E)$ is isometrically isomorphic to the product space of the asymptotic Teichmüller spaces of the connected components of $ \hat {\mathbb{C}}\setminus E$ and the Banach space of the Beltrami coefficients defined on $ E$. Furthermore, it is proved that there is a complex Banach manifold structure on $ AT(E)$.


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Additional Information

Yi Qi
Affiliation: Key Laboratory of Ministry of Education – “Mathematics, Informatics and Behavioral Semantics”, School of Mathematics and Systems Science, Beihang University, Beijing 100191, People’s Republic of China
Email: yiqi@buaa.edu.cn

Yan Wu
Affiliation: Key Laboratory of Ministry of Education – “Mathematics, Informatics and Behavioral Semantics”, School of Mathematics and Systems Science, Beihang University, Beijing 100191, People’s Republic of China
Address at time of publication: School of Science, Linyi University, Shandong 276005, People’s Republic of China
Email: by1209113@buaa.edu.cn

DOI: https://doi.org/10.1090/proc/13144
Keywords: Teichm\"uller space, Teichm\"uller space of a closed set, asymptotic Teichm\"uller space of a closed set, quasiconformal mapping
Received by editor(s): July 30, 2014
Received by editor(s) in revised form: January 26, 2016
Published electronically: March 14, 2018
Additional Notes: This research was partially supported by the National Natural Science Foundation of China (Grant Nos.11371045, 11701250) and the Fundamental Research Funds for the Central University.
Yan Wu is the corresponding author.
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2018 American Mathematical Society

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