The modulus of the Korányi ellipsoidal ring
HTML articles powered by AMS MathViewer
- by Gaoshun Gou and Ioannis D. Platis PDF
- Proc. Amer. Math. Soc. 147 (2019), 2975-2986 Request permission
Abstract:
The Korányi ellipsoidal ring $\mathcal {E}=\mathcal {E}_{B,A}$, $0<B<A$, is defined as the image of the Korányi spherical ring centred at the origin and of radii $B$ and $A$ via a linear contact quasiconformal map $L$ in the Heisenberg group. If $K\ge 1$ is the maximal distortion of $L$, then we prove that the modulus of $\mathcal {E}$ is equal to \begin{equation*} \textrm {mod}(\mathcal {E})=\left (\frac {3}{8}\Big (K^2+\frac {1}{K^2}\Big )+\frac {1}{4}\right )\frac {\pi ^2}{(\log (A/B))^3}. \end{equation*}References
- Zoltán M. Balogh, Hausdorff dimension distribution of quasiconformal mappings on the Heisenberg group, J. Anal. Math. 83 (2001), 289–312. MR 1828495, DOI 10.1007/BF02790265
- Zoltán M. Balogh, Katrin Fässler, and Ioannis D. Platis, Modulus method and radial stretch map in the Heisenberg group, Ann. Acad. Sci. Fenn. Math. 38 (2013), no. 1, 149–180. MR 3076803, DOI 10.5186/aasfm.2013.3811
- Jin Fa Cheng and Yu Ming Chu, Extremal mappings of elliptic regions under prescribed conditions and their properties, J. Shanghai Jiaotong Univ. 29 (1995), no. 5, 136–143 (Chinese, with English and Chinese summaries). MR 1385854
- A. Korányi and H. M. Reimann, Quasiconformal mappings on the Heisenberg group, Invent. Math. 80 (1985), no. 2, 309–338. MR 788413, DOI 10.1007/BF01388609
- Adam Korányi and Hans Martin Reimann, Horizontal normal vectors and conformal capacity of spherical rings in the Heisenberg group, Bull. Sci. Math. (2) 111 (1987), no. 1, 3–21 (English, with French summary). MR 886958
- A. Korányi and H. M. Reimann, Foundations for the theory of quasiconformal mappings on the Heisenberg group, Adv. Math. 111 (1995), no. 1, 1–87. MR 1317384, DOI 10.1006/aima.1995.1017
- Juha Heinonen and Pekka Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), no. 1, 1–61. MR 1654771, DOI 10.1007/BF02392747
- Pierre Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), no. 1, 1–60 (French, with English summary). MR 979599, DOI 10.2307/1971484
- Ioannis D. Platis, Modulus of revolution rings in the Heisenberg group, Proc. Amer. Math. Soc. 144 (2016), no. 9, 3975–3990. MR 3513553, DOI 10.1090/proc/13060
- Edgar Reich, On the structure of the family of extremal quasiconformal mappings of parabolic regions, Complex Variables Theory Appl. 5 (1986), no. 2-4, 289–300. MR 846497, DOI 10.1080/17476938608814149
Additional Information
- Gaoshun Gou
- Affiliation: Department of Mathematics, Hunan University, Changsha 410082, People’s Republic of China
- Email: gaoshungou@hnu.edu.cn
- Ioannis D. Platis
- Affiliation: Department of Mathematics and Applied Mathematics, University of Crete, Heraklion Crete 70013, Greece
- MR Author ID: 659998
- ORCID: 0000-0002-0656-0856
- Email: jplatis@math.uoc.gr
- Received by editor(s): August 8, 2018
- Received by editor(s) in revised form: October 8, 2018
- Published electronically: March 7, 2019
- Additional Notes: The first author was funded by NSFC No. 11631010 and NSFC No. 11701165 grants.
- Communicated by: Jeremy Tyson
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 2975-2986
- MSC (2010): Primary 30L10, 30C75
- DOI: https://doi.org/10.1090/proc/14434
- MathSciNet review: 3973899