Modulus of revolution rings in the heisenberg group

Abstract:

Let $\mathcal {S}$ be a surface of revolution embedded in the Heisenberg group $\mathfrak {H}$. A revolution ring $R_{a,b}(\mathcal {S})$, $0<a<b$, is a domain in $\mathfrak {H}$ bounded by two dilated images of $\mathcal {S}$, with dilation factors $a$ and $b$, respectively. We prove that if $\mathcal {S}$ is subject to certain geometric conditions, then the modulus of the family $\Gamma$ of horizontal boundary connecting curves inside $R_{a,b}(\mathcal {S})$ is \[ \textrm {Mod}(\Gamma )=\pi ^2(\log (b/a))^{-3}. \] Our result applies for many interesting surfaces, e.g., the Korányi metric sphere, the Carnot-Carathéodory metric sphere and the bubble set.