Refereed journal article or data article (A1)
Teichmüller's problem in space
List of Authors: Klen R, Todorcevic V, Vuorinen M
Publisher: ACADEMIC PRESS INC ELSEVIER SCIENCE
Publication year: 2017
Journal: Journal of Mathematical Analysis and Applications
Journal name in source: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
Journal acronym: J MATH ANAL APPL
Volume number: 455
Issue number: 1-2
Start page: 1297
End page: 1316
Number of pages: 20
ISSN: 0022-247X
DOI: http://dx.doi.org/10.1016/j.jmaa.2017.06.026
Abstract
Quasiconformal homeomorphisms of the whole space, 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 Teichmuller. Unlike Teichmuller'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 R-n {0} onto itself. (C) 2017 Elsevier Inc. All rights reserved.
Quasiconformal homeomorphisms of the whole space, 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 Teichmuller. Unlike Teichmuller'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 R-n {0} onto itself. (C) 2017 Elsevier Inc. All rights reserved.
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