Asymptotic Teichmüller space of a closed set of the Riemann sphere

Qi, Yi; Wu, Yan

doi:10.1090/proc/13144

Asymptotic Teichmüller space of a closed set of the Riemann sphere
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by Yi Qi and Yan Wu

Proc. Amer. Math. Soc. 146 (2018), 2867-2876

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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