Modulus of revolution rings in the Heisenberg group

TL;DR

This paper derives an explicit formula for the modulus of certain families of curves within revolution rings in the Heisenberg group, extending understanding of geometric function theory in sub-Riemannian spaces.

Contribution

It provides a new explicit formula for the modulus of boundary connecting curves in revolution rings in the Heisenberg group under specific geometric conditions.

Findings

The modulus formula is ( b/a)^{-3} for revolution rings.

Applicable to spheres in Kore1nyi and Carnot-Carathe9odory metrics.

Includes analysis of the bubble set in the Heisenberg group.

Abstract

Let $\mathcal{S}$ be a surface of revolution embedded in the Heisenberg group $\mathcal{H}$. A revolution ring $R_{a,b}(\mathcal{S})$, $0<a<b$, is a domain in $\mathcal{H}$ 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 $$ {\rm Mod}(\Gamma)=\pi^2(\log(b/a))^{-3}. $$ Our result applies for many interesting surfaces, e.g., the Kor\'anyi metric sphere, the Carnot-Carath\'eodory metric sphere and the bubble set.