Authors: Klén, Riku
Todorčević, Vesna 
Vuorinen, Mattiv
Title: Teichmüller's problem in space
Journal: Journal of Mathematical Analysis and Applications
Volume: 455
Issue: 2
First page: 1297
Last page: 1316
Issue Date: 15-Nov-2017
Rank: M21
ISSN: 0022-247X
DOI: 10.1016/j.jmaa.2017.06.026
Quasiconformal homeomorphisms of the whole space Rn, onto itself normalized at one or two points are studied. In particular, the stability theory, the case when the maximal dilatation tends to 1, is in the focus. Our main result provides a spatial analogue of a classical result due to Teichmüller. Unlike Teichmüller's result, our bounds are explicit. Explicit bounds are based on two sharp well-known distortion results: the quasiconformal Schwarz lemma and the bound for linear dilatation. Moreover, Bernoulli type inequalities and asymptotically sharp bounds for special functions involving complete elliptic integrals are applied to simplify the computations. Finally, we discuss the behavior of the quasihyperbolic metric under quasiconformal maps and prove a sharp result for quasiconformal maps of Rn∖{0} onto itself.
Keywords: Distance-ratio metric | Quasiconformal mappings | Quasihyperbolic metric
Publisher: Elsevier

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