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Modulus of surface families and the radial stretch in the Heisenberg group
Published online by Cambridge University Press: 26 May 2016
Abstract
We develop a modulus method for surface families inside a domain in the Heisenberg group and we prove that the stretch map between two Heisenberg spherical rings is a minimiser for the mean distortion among the class of contact quasiconformal maps between these rings which satisfy certain boundary conditions.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 162 , Issue 1 , January 2017 , pp. 13 - 37
- Copyright
- Copyright © Cambridge Philosophical Society 2016
References
REFERENCES
[1]
Astala, K., Iwaniec, T., Martin, G. J. and Onninen, J.
Extremal mappings of finite distortion. Proc. Lon. Math. Soc.
3 (2005), 655–702.Google Scholar
[2]
Balogh, Z. M., Fässler, K. and Platis, I. D.
Modulus method and radial stretch map in the Heisenberg group. Ann. Acad. Sci. Fenn.
38 (2013), 1–32.Google Scholar
[3]
Balogh, Z. M., Fässler, K. and Platis, I. D.
Modulus of curve families and extremality of spiral–stretch maps. J. Anal. Math.
113 (2011), 265–291.Google Scholar
[4]
Capogna, L., Danielli, D., Pauls, S. D. and Tyson, J. T.
An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem. Progr. Math.
259 (Birkhuser Verlag, Basel, 2007).Google Scholar
[5]
Dairbekov, N. S.
On mappings of bounded distortion on the Heisenberg group. Sib. Math. Zh.
41
(1), (2000), 49–59.Google Scholar
[6]
Do Carmo, M. P. Differential geometry of curves and surfaces (Prentice-Hall, NJ, 1976).Google Scholar
[7]
Fuglede, B.
Extremal length and functional completion. Acta Math.
98
(1), 171–219.Google Scholar
[8]
Gardiner, F. and Lakic, N.
Quasiconformal Teichmüller theory. Math. Surv. and Mon.
76 (AMS, 2000).Google Scholar
[9]
Grötzsch, H.
Über möglichst konforme Abbildungen von schlichten Bereichen. Ber. Math.-phys. Kl. Sachs. Akad. Wis. Leipsig.
84 (1932), 114–120.Google Scholar
[10]
Gutlyanskii, V. and Martio, O.
Rotation estimates and spirals. Conf. Geom. Dyn.
5 (2001), 6–20.Google Scholar
[11]
Heinonen, J. Calculus on Carnot groups. In: Fall school in Analysis (Jyväskylä, 1994), Report 68 (Univ. Jyväskylä, Jyväskylä, 1995), 1–31.Google Scholar
[12]
Kirchheim, B. and Serra Cassano, F.
Rectifiability and parametrization of intrinsic regular surfaces in the Heisenberg group. Ann. Sc. Norm. Sup. Pisa CI. Sci.
5 (2004), 871–896.Google Scholar
[13]
Korányi, A. and Reimann, H. M.
Foundations for the theory of quasiconformal mappings of the Heisenberg group. Adv. in Math.
111 (1995), 1–87.Google Scholar
[14]
Korányi, A. and Reimann, H. M.
Quasiconformal mappings on the Heisenberg group. Invent. Math.
80
(2) (1985), 309–338.Google Scholar
[15]
Martin, G. J.
The Teichmüller problem for mean distortion. Ann. Acad. Sci. Fenn. Math.
34 (2009), 233–247.Google Scholar
[16]
Mostow, G. D.
Strong rigidity in locally symmetric spaces. Ann. Math. Stud.
78 (Princeton University Press, Princeton, N.J., 1973).Google Scholar
[17]
Pansu, P.
Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. of Math.
129 (1989), 1–60.CrossRefGoogle Scholar
[18]
Platis, I. D.
Straight ruled surfaces in the Heisenberg group. J. Geom.
105 (2014), 119–138.CrossRefGoogle Scholar
[19]
Platis, I. D.
The geometry of complex hyperbolic packs. Math. Proc. Cam. Phil.
147 (2009), 205–234.Google Scholar
[21]
Strebel, K.
Quadratic differentials (Springer-Verlag, Berlin and New York, 1984).Google Scholar
[22]
Teichmüller, O.
Extremale quasikonforme Abbildungen und quadratische Differentiale. Abh. Preuss. Akad. Wiss. Math.-Nat. Kl.
22 (1939), 197.Google Scholar
[23]
Vasil'ev, A.
Moduli of families of curves for conformal and quasiconformal mappings (Springer-Verlag, Berlin and New York, 2004).Google Scholar
[24]
Vodop'yanov, S. K.
Monotone functions and quasiconformal mappings on Carnot groups. Sib. Math. Zh.
37
(6) (1996), 1269–1295.Google Scholar