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The modulus method introduced by H. Grötzsch yields bounds for a mean distortion functional of quasiconformal maps between two annuli mapping the respective boundary components onto each other. P. P. Belinskiĭ studied these inequalities in the plane and identified the family of all minimisers. Beyon...

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Bibliographic Details
Published inConformal geometry and dynamics Vol. 19; no. 6; pp. 122 - 145
Main Authors Zoltán M. Balogh, Katrin Fässler, Ioannis D. Platis
Format Journal Article
LanguageEnglish
Published 06.05.2015
Subjects
Heisenberg group
modulus method
mean distortion
extremal quasiconformal mappings
uniqueness of minimisers
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Summary:The modulus method introduced by H. Grötzsch yields bounds for a mean distortion functional of quasiconformal maps between two annuli mapping the respective boundary components onto each other. P. P. Belinskiĭ studied these inequalities in the plane and identified the family of all minimisers. Beyond the Euclidean framework, a Grötzsch-Belinskiĭ-type inequality has been previously considered for quasiconformal maps between annuli in the Heisenberg group whose boundaries are Korányi spheres. In this note we show that--in contrast to the planar situation--the minimiser in this setting is essentially unique.
ISSN:1088-4173
DOI:10.1090/ecgd/278