%0 Journal Article
%T On the lack of density of Lipschitz mappings in Sobolev spaces with Heisenberg target
%A Noel DeJarnette
%A Piotr Hajlasz
%A Anton Lukyanenko
%A Jeremy Tyson
%J Mathematics
%D 2011
%I arXiv
%X We study the question: when are Lipschitz mappings dense in the Sobolev space $W^{1,p}(M,\mathbf{H}^n)$? Here $M$ denotes a compact Riemannian manifold with or without boundary, while $\mathbf{H}^n$ denotes the $n$th Heisenberg group equipped with a sub-Riemannian metric. We show that Lipschitz maps are dense in $W^{1,p}(M,\mathbf{H}^n)$ for all $1\le p<\infty$ if $\dim M \le n$, but that Lipschitz maps are not dense in $W^{1,p}(M,\mathbf{H}^n)$ if $\dim M \ge n+1$ and $n\le p<n+1$. The proofs rely on the construction of smooth horizontal embeddings of the sphere $S^n$ into $\mathbf{H}^n$. We provide two such constructions, one arising from complex hyperbolic geometry and the other arising from symplectic geometry. The nondensity assertion can be interpreted as nontriviality of the $n$th Lipschitz homotopy group of $\mathbf{H}^n$. We initiate a study of Lipschitz homotopy groups for sub-Riemannian spaces.
%U http://arxiv.org/abs/1109.4641v3