Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Modulus of revolution rings in the Heisenberg group   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.

29
The categories of this paper are: funct-an
33
The categories of this paper are: plasm-ph
9
The categories of this paper are: nlin
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
16
The categories of this paper are: stat
6
The categories of this paper are: astro-ph
13
The categories of this paper are: quant-ph
24
The categories of this paper are: dg-ga
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The categories of this paper are: mtrl-th
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The categories of this paper are: cmp-lg
17
The categories of this paper are: q-fin
3
The categories of this paper are: physics
1
The categories of this paper are: cs
20
The categories of this paper are: acc-phys
5
The categories of this paper are: gr-qc