Abstract
We prove that a self-homeomorphism of the Grushin plane is quasisymmetric if and only if it is metrically quasiconformal and if and only if it is geometrically quasiconformal. As the main step in our argument, we show that a quasisymmetric parametrization of the Grushin plane by the Euclidean plane must also be geometrically quasiconformal. We also discuss some aspects of the Euclidean theory of quasiconformal maps, such as absolute continuity on almost every compact curve, not satisfied in the Grushin case.
Similar content being viewed by others
References
Ackermann, C.: An approach to studying quasiconformal mappings on generalized Grushin planes. Ann. Acad. Sci. Fenn. 40(1), 305–320 (2015). http://www.acadsci.fi/mathematica/Vol40/Ackermann.html
Bellaïche, A.: The tangent space in sub-Riemannian geometry. In: Sub-Riemannian geometry, Progr. Math., vol. 144, pp. 1–78. Birkhäuser, Basel (1996). doi:10.1007/978-3-0348-9210-0_1
Heinonen, J.: Lectures on analysis on metric spaces. Universitext. Springer, New York (2001). doi:10.1007/978-1-4613-0131-8
Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.T.: Sobolev classes of Banach space-valued functions and quasiconformal mappings. J. Anal. Math. 85, 87–139 (2001). doi:10.1007/BF02788076
Korányi, A., Reimann, H.M.: Foundations for the theory of quasiconformal mappings on the Heisenberg group. Adv. Math. 111(1), 1–87 (1995). doi:10.1006/aima.1995.1017
Lehto, O., Virtanen, K.I., Väisälä, J.: Contributions to the distortion theory of quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A I 273, 14 (1959)
Margulis, G.A., Mostow, G.D.: The differential of a quasi-conformal mapping of a Carnot–Carathéodory space. Geom. Funct. Anal. 5(2), 402–433 (1995). doi:10.1007/BF01895673
Meyerson, W.: The Grushin plane and quasiconformal Jacobians. arXiv preprint arXiv:1112.0078 (2011). http://arxiv.org/abs/1112.0078
Monti, R., Morbidelli, D.: Isoperimetric inequality in the Grushin plane. J. Geom. Anal. 14(2), 355–368 (2004). doi:10.1007/BF02922077
Morbidelli, D.: Liouville theorem, conformally invariant cones and umbilical surfaces for Grushin-type metrics. Israel J. Math. 173, 379–402 (2009). doi:10.1007/s11856-009-0097-7
Payne, K.R.: Singular metrics and associated conformal groups underlying differential operators of mixed and degenerate types. Ann. Mat. Pura Appl. 185(4), 613–625 (2006). doi:10.1007/s10231-005-0173-5
Rajala, K.: Uniformization of two-dimensional metric surfaces. Invent. Math. (2016). doi:10.1007/s00222-016-0686-0
Rickman, S.: Quasiregular Mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26. Springer, Berlin (1993). doi:10.1007/978-3-642-78201-5
Romney, M.: Conformal Grushin spaces. Conform. Geom. Dyn. 20, 97–115 (2016). doi:10.1090/ecgd/292
Romney, M., Vellis, V.: Bi-Lipschitz embedding of the generalized Grushin plane in Euclidean spaces. Math. Res. Lett. (to appear)
Väisälä, J.: Lectures on \(n\)-dimensional quasiconformal mappings. Lecture Notes in Mathematics, vol. 229. Springer, Berlin (1971)
Vuorinen, M.: Quadruples and spatial quasiconformal mappings. Math. Z. 205, 617–628 (1990). doi:10.1007/BF02571267
Wu, J.M.: Geometry of Grushin spaces. Illinois J. Math. 59(1), 21–41 (2015). https://projecteuclid.org/euclid.ijm/1455203157
Acknowledgements
The authors thank Jeremy Tyson and Colleen Ackermann for comments on a draft of this paper. They also thank the referee for useful feedback.
Author information
Authors and Affiliations
Corresponding author
Additional information
D. Jung was supported by U.S. Department of Education GAANN fellowship P200A150319.
Rights and permissions
About this article
Cite this article
Gartland, C., Jung, D. & Romney, M. Quasiconformal mappings on the Grushin plane. Math. Z. 287, 915–928 (2017). https://doi.org/10.1007/s00209-017-1851-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-017-1851-x