Abstract
Denote by \({\mathbb{H}^n}\) the 2n + 1 dimensional Heisenberg group. We show that the pairs \({(\mathbb{R}^k ,\mathbb{H}^n)}\) and \({(\mathbb{H}^k ,\mathbb{H}^n)}\) do not have the Lipschitz extension property for k > n.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allcock D.: An isoperimetric inequality for the Heisenberg groups. Geom. Funct. Anal. 8(2), 219–233 (1998)
Ambrosio L., Kirchheim B.: Rectifiable sets in metric and Banach spaces. Math. Ann. 318(3), 527–555 (2000)
Assouad P.: Sur la distance de Nagata. C. R. Acad. Sci. Paris Sér. I Math. 294(1), 31–34 (1982)
Balogh Z.M., Rickly M., Serra Cassano F.: Comparison of Hausdorff measures with respect to the Euclidean and the Heisenberg metric. Publ. Mat. 47(1), 237–259 (2003)
Balogh Z.M., Tyson J.T., Warhurst B.: Gromov’s dimension comparison problem on Carnot groups. C. R. Math. Acad. Sci. Paris 346(3–4), 135–138 (2008)
Balogh, Z.M., Tyson, J.T., Warhurst, B.: Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups. Adv. Math. (2008, to appear)
Brudnyi A., Brudnyi Y.: Metric spaces with linear extensions preserving Lipschitz condition. Am. J. Math. 129(1), 217–314 (2007)
Ekholm T., Etnyre J., Sullivan M.: Non-isotopic Legendrian submanifolds in \({\mathbb{R}^{2n+1}}\). J. Differential Geom. 71(1), 85–128 (2005)
Fässler, K.: Extending Lipschitz maps from Euclidean spaces into Heisenberg groups. Master’s thesis, Mathematisches Institut, Universität Bern (2007)
Franchi B., Serapioni R., Serra Cassano F.: Regular submanifolds, graphs and area formula in Heisenberg groups. Adv. Math. 211(1), 152–203 (2007)
Gromov, M.: Carnot-Carathéodory spaces seen from within. In: Sub-Riemannian geometry, Progr. Math., vol. 144, pp. 79–323. Birkhäuser, Basel (1996)
Hajłasz, P., Tyson, J.T.: Density of Lipschitz maps in Sobolev spaces with Heisenberg group target (2008, in preparation)
Hajłasz, P., Tyson, J.T.: Lipschitz, Holder and Sobolev surjections from Euclidean cubes (2008, in preparation)
Johnson W.B., Lindenstrauss J., Schechtman G.: Extensions of Lipschitz maps into Banach spaces. Israel J. Math. 54(2), 129–138 (1986)
Kirchheim B., Serra Cassano F.: Rectifiability and parameterization of intrinsic regular surfaces in the Heisenberg group. Ann. Sci. Norm. Super. Pisa Cl. Sci. (5) 3(4), 871–896 (2004)
Lang U., Schlichenmaier T.: Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions. Int. Math. Res. Not 58, 3625–3655 (2005)
Lee J.R., Naor A.: Extending Lipschitz functions via random metric partitions. Invent. Math. 160(1), 59–95 (2005)
Magnani V.: Unrectifiability and rigidity in stratified groups. Arch. Math. (Basel) 83(6), 568–576 (2004)
Magnani, V.: Contact equations and Lipschitz extensions (2007). ArXiv:0711.5003v1
Mattila, P.: Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge University Press, Cambridge (1995)
Milnor, J.W.: Topology from the differentiable viewpoint. Princeton Landmarks in Mathematics. Princeton University Press (1997). Based on notes by David W. Weaver; Revised reprint of the 1965 original
Wells, J.H., Williams, L.R.: Embeddings and extensions in analysis. Springer-Verlag, New York (1975). Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors were supported by Swiss National Science Foundation, European Research Council Project GALA and European Science Foundation Project HCAA.
Rights and permissions
About this article
Cite this article
Balogh, Z.M., Fässler, K.S. Rectifiability and Lipschitz extensions into the Heisenberg group. Math. Z. 263, 673–683 (2009). https://doi.org/10.1007/s00209-008-0437-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-008-0437-z