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,Hⁿ)? Here M denotes a compact Riemannian manifold with or without boundary, while Hⁿ denotes the nth Heisenberg group equipped with a sub-Riemannian metric. We show that Lipschitz maps are dense in W^1,p(M,Hⁿ) for all 1 ≤ p < ∞ if dimM ≤ n, but that Lipschitz maps are not dense in W^1,p(M,Hⁿ) if dimM ≥ n+1 and n ≤ p < n+1. The proofs rely on the construction of smooth horizontal embeddings of the sphere Sⁿ into Hⁿ. 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 nth Lipschitz homotopy group of Hⁿ. We initiate a study of Lipschitz homotopy groups for sub-Riemannian spaces.