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Modulus of revolution rings in the heisenberg group


Author: Ioannis D. Platis
Journal: Proc. Amer. Math. Soc. 144 (2016), 3975-3990
MSC (2010): Primary 30L05, 30C75
DOI: https://doi.org/10.1090/proc/13060
Published electronically: March 17, 2016
MathSciNet review: 3513553
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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

$\displaystyle {\rm 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.

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Additional Information

Ioannis D. Platis
Affiliation: Department of Mathematics and Applied Mathematics, University of Crete, University Campus, GR 700 13 Voutes Heraklion Crete, Greece
Email: jplatis@math.uoc.gr

DOI: https://doi.org/10.1090/proc/13060
Keywords: Heisenberg group, surfaces of revolution, modulus of curve families
Received by editor(s): June 2, 2015
Received by editor(s) in revised form: October 12, 2015, November 5, 2015, and November 19, 2015
Published electronically: March 17, 2016
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2016 American Mathematical Society