Tohoku Mathematical Journal

Teichmüller spaces and tame quasiconformal motions

Yunping Jiang, Sudeb Mitra, Hiroshige Shiga, and Zhe Wang

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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.

Article information

Source
Tohoku Math. J. (2), Volume 70, Number 4 (2018), 607-631.

Dates
First available in Project Euclid: 4 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1546570827

Digital Object Identifier
doi:10.2748/tmj/1546570827

Mathematical Reviews number (MathSciNet)
MR3896139

Zentralblatt MATH identifier
07040978

Subjects
Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]
Secondary: 30C99: None of the above, but in this section 30F99: None of the above, but in this section 37F30: Quasiconformal methods and Teichmüller theory; Fuchsian and Kleinian groups as dynamical systems

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

Citation

Jiang, Yunping; Mitra, Sudeb; Shiga, Hiroshige; Wang, Zhe. Teichmüller spaces and tame quasiconformal motions. Tohoku Math. J. (2) 70 (2018), no. 4, 607--631. doi:10.2748/tmj/1546570827. https://projecteuclid.org/euclid.tmj/1546570827


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