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 W 1 , p ( M , H n ) W^{1,p}(M,\mathbb {H}^n) ? Here M M denotes a compact Riemannian manifold with or without boundary, while H n \mathbb {H}^n denotes the n n th Heisenberg group equipped with a sub-Riemannian metric. We show that Lipschitz maps are dense in W 1 , p ( M , H n ) W^{1,p}(M,\mathbb {H}^n) for all 1 ≤ p > ∞ 1\le p>\infty if dim ⁡ M ≤ n \dim M \le n , but that Lipschitz maps are not dense in W 1 , p ( M , H n ) W^{1,p}(M,\mathbb {H}^n) if dim ⁡ M ≥ n + 1 \dim M \ge n+1 and n ≤ p > n + 1 n\le p>n+1 . The proofs rely on the construction of smooth horizontal embeddings of the sphere S n \mathbb {S}^n into H n \mathbb {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 n th Lipschitz homotopy group of H n \mathbb {H}^n . We initiate a study of Lipschitz homotopy groups for sub-Riemannian spaces.