On the lack of density of lipschitz mappings in sobolev spaces with heisenberg target

Abstract

We study the question: When are Lipschitz mappings dense in the Sobolev space W1,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 W1,p(M,Hn) for all 1 ≤ p < ∞ if dimM ≤ n, but that Lipschitz maps are not dense in W1,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.