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On the lack of density of Lipschitz mappings in Sobolev spaces with Heisenberg target

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.