The modulus of the Korányi ellipsoidal ring

Abstract

The Korányi ellipsoidal ring $\mathcal{E}$ of radii $B$ and $A$, $0<B<A$, is defined as the image of the Korányi 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 $$\mathrm{mod}(\mathcal{E})=\left(\frac{3}{8}(K^2+\frac{1}{K^2})+\frac{1}{4}\right)\frac{{\pi }^2}{(\log (A/B){)}^3}.$$

10 pages