The modulus of the Kor\'anyi ellipsoidal ring

TL;DR

This paper derives an explicit formula for the modulus of Korányi ellipsoidal rings in the Heisenberg group, relating it to the maximal distortion of linear contact maps.

Contribution

It provides the first explicit computation of the modulus for Korányi ellipsoidal rings under linear contact maps in the Heisenberg group.

Findings

The modulus formula depends on the maximal distortion K.

The modulus is proportional to rac{ig(K^2 + rac{1}{K^2}ig)}{( ext{log}(A/B))^3}.

The result generalizes known cases for spherical rings.

Abstract

The Kor\'anyi ellipsoidal ring $\mathcal{E}$ of radii $B$ and $A$, $0<B<A$, is defined as the image of the Kor\'anyi spherical ring of the same radii and centred at the origin via a linear contact 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 $$ {\rm mod}(\mathcal{E})=\left(\frac{3}{8}\Big(K^2+\frac{1}{K^2}\Big)+\frac{1}{4}\right)\frac{\pi^2}{(\log (A/B))^3}. $$