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NONDECREASABLE AND WEAKLY NONDECREASABLE DILATATIONS

Part of: Geometric function theory Deformations of analytic structures Riemann surfaces

Published online by Cambridge University Press:  19 September 2016

GUOWU YAO
GUOWU YAO*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR China email gwyao@math.tsinghua.edu.cn
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Abstract

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Zhou et al. [‘On weakly non-decreasable quasiconformal mappings’, J. Math. Anal. Appl.386 (2012), 842–847] proved that, in a Teichmüller equivalence class, there exists an extremal quasiconformal mapping with a weakly nondecreasable dilatation. They asked whether a weakly nondecreasable dilatation is a nondecreasable dilatation. The aim of this paper is to give a negative answer to their problem. We also construct a Teichmüller class such that it contains an infinite number of weakly nondecreasable extremal representatives, only one of which is nondecreasable.

Keywords

Teichmüller spacequasiconformal mappingweakly nondecreasablepseudo nondecreasable

MSC classification

Primary: 30C75: Extremal problems for conformal and quasiconformal mappings, other methods 30C62: Quasiconformal mappings in the plane
Secondary: 30F60: Teichmüller theory 32G15: Moduli of Riemann surfaces, Teichmüller theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 102 , Issue 3 , June 2017 , pp. 420 - 434
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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