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Uniqueness of minimisers for a Grötzsch-Belinskiĭ type inequality in the Heisenberg group


Authors: Zoltán M. Balogh, Katrin Fässler and Ioannis D. Platis
Journal: Conform. Geom. Dyn. 19 (2015), 122-145
MSC (2010): Primary 30L10, 30C75
DOI: https://doi.org/10.1090/ecgd/278
Published electronically: May 6, 2015
MathSciNet review: 3343051
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Abstract: 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.


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Additional Information

Zoltán M. Balogh
Affiliation: Mathematisches Institut, Sidlerstrasse 5, CH-3012 Bern, Switzerland
Email: zoltan.balogh@math.unibe.ch

Katrin Fässler
Affiliation: Mathematisches Institut, Sidlerstrasse 5, CH-3012 Bern, Switzerland
Email: katrin.faessler@math.unibe.ch

Ioannis D. Platis
Affiliation: Department of Mathematics and Applied Mathematics, University of Crete, University Campus, GR-70013 Heraklion Crete, Greece
Email: jplatis@math.uoc.gr

DOI: https://doi.org/10.1090/ecgd/278
Keywords: Heisenberg group, extremal quasiconformal mappings, modulus method, mean distortion, uniqueness of minimisers
Received by editor(s): November 7, 2014
Received by editor(s) in revised form: March 31, 2015
Published electronically: May 6, 2015
Additional Notes: This research was supported by the Swiss National Science Foundaton
Article copyright: © Copyright 2015 American Mathematical Society