Modulus of revolution rings in the heisenberg group
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- by Ioannis D. Platis PDF
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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 \[ \textrm {Mod}(\Gamma )=\pi ^2(\log (b/a))^{-3}. \] Our result applies for many interesting surfaces, e.g., the Korányi metric sphere, the Carnot-Carathéodory metric sphere and the bubble set.References
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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
- MR Author ID: 659998
- ORCID: 0000-0002-0656-0856
- Email: jplatis@math.uoc.gr
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 3975-3990
- MSC (2010): Primary 30L05, 30C75
- DOI: https://doi.org/10.1090/proc/13060
- MathSciNet review: 3513553