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The modulus of the Korányi ellipsoidal ring

Authors: Gaoshun Gou and Ioannis D. Platis
Journal: Proc. Amer. Math. Soc. 147 (2019), 2975-2986
MSC (2010): Primary 30L10, 30C75
Published electronically: March 7, 2019
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Abstract: The Korányi ellipsoidal ring $ \mathcal {E}=\mathcal {E}_{B,A}$, $ 0<B<A$, is defined as the image of the Korányi spherical ring centred at the origin and of radii $ B$ and $ A$ via a linear contact quasiconformal map $ L$ in the Heisenberg group. If $ K\ge 1$ is the maximal distortion of $ L$, then we prove that the modulus of $ \mathcal {E}$ is equal to

$\displaystyle {\rm mod}(\mathcal {E})=\left (\frac {3}{8}\Big (K^2+\frac {1}{K^2}\Big )+\frac {1}{4}\right )\frac {\pi ^2}{(\log (A/B))^3}.$    

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Additional Information

Gaoshun Gou
Affiliation: Department of Mathematics, Hunan University, Changsha 410082, People’s Republic of China

Ioannis D. Platis
Affiliation: Department of Mathematics and Applied Mathematics, University of Crete, Heraklion Crete 70013, Greece

Keywords: Kor\'anyi ellipsoidal rings, Kor\'anyi spherical rings, modulus, conformal capacity, Heisenberg group.
Received by editor(s): August 8, 2018
Received by editor(s) in revised form: October 8, 2018
Published electronically: March 7, 2019
Additional Notes: The first author was funded by NSFC No. 11631010 and NSFC No. 11701165 grants.
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2019 American Mathematical Society