Tohoku Mathematical Journal
2018
December
SECOND SERIES VOL. 70, NO. 4

Tohoku Math. J.
70 (2018), 607-631

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.

To the top of this page

Back to the Contents