The modulus of the Korányi ellipsoidal ring

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 \begin{equation*} \textrm {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}. \end{equation*}