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HOME > Table of Contents and Abstracts > Vol. 70, No. 4
Tohoku Mathematical Journal
2018
December
SECOND SERIES VOL. 70, NO. 4
Tohoku Math. J.
70 (2018), 607-631
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Title
TEICHMÜLLER SPACES AND TAME QUASICONFORMAL MOTIONS
Author
Yunping Jiang, Sudeb Mitra, Hiroshige Shiga and Zhe Wang
(Received September 9, 2015, revised July 28, 2016) |
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.
Mathematics Subject Classification.
Primary 32G15, Secondary 30C99, 30F99, 37F30.
Key words and phrases.
Quasiconformal motions, tame quasiconformal motions, holomorphic motions, continuous motions, Teichmüller spaces.
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