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

Balogh, Zoltán; Fässler, Katrin; Platis, Ioannis

doi:10.1090/ecgd/278

Uniqueness of minimisers for a Grötzsch-Belinskiĭ type inequality in the Heisenberg group
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by Zoltán M. Balogh, Katrin Fässler and Ioannis D. Platis

Conform. Geom. Dyn. 19 (2015), 122-145

DOI: https://doi.org/10.1090/ecgd/278

Published electronically: May 6, 2015

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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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