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2018 Teichmüller spaces and tame quasiconformal motions
Yunping Jiang, Sudeb Mitra, Hiroshige Shiga, Zhe Wang
Tohoku Math. J. (2) 70(4): 607-631 (2018). DOI: 10.2748/tmj/1546570827

Abstract

The concept of “quasiconformal motion” was first introduced by Sullivan and Thurston (in [24]). Theorem 3 of that paper asserted that any quasiconformal motion of a set in the sphere over an interval can be extended to the sphere. In this paper, we give a counter-example to that assertion. We introduce a new concept called “tame quasiconformal motion” and show that their assertion is true for tame quasiconformal motions. We prove a much more general result that, any tame quasiconformal motion of a closed set in the sphere, over a simply connected Hausdorff space, can be extended as a quasiconformal motion of the sphere. Furthermore, we show that this extension can be done in a conformally natural way. The fundamental idea is to show that the Teichmüller space of a closed set in the sphere is a “universal parameter space” for tame quasiconformal motions of that set over a simply connected Hausdorff space.

Citation

Download Citation

Yunping Jiang. Sudeb Mitra. Hiroshige Shiga. Zhe Wang. "Teichmüller spaces and tame quasiconformal motions." Tohoku Math. J. (2) 70 (4) 607 - 631, 2018. https://doi.org/10.2748/tmj/1546570827

Information

Published: 2018
First available in Project Euclid: 4 January 2019

zbMATH: 07040978
MathSciNet: MR3896139
Digital Object Identifier: 10.2748/tmj/1546570827

Subjects:
Primary: 32G15
Secondary: 30C99 , 30F99 , 37F30

Keywords: continuous motions , holomorphic motions , quasiconformal motions , tame quasiconformal motions , Teichmüller spaces

Rights: Copyright © 2018 Tohoku University

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Vol.70 • No. 4 • 2018
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