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 $W^{1,p}(M,\mathbb {H}^n)$? Here $M$ denotes a compact Riemannian manifold with or without boundary, while $\mathbb {H}^n$ denotes the $n$th Heisenberg group equipped with a sub-Riemannian metric. We show that Lipschitz maps are dense in $W^{1,p}(M,\mathbb {H}^n)$ for all $1\le p<\infty$ if $\dim M \le n$, but that Lipschitz maps are not dense in $W^{1,p}(M,\mathbb {H}^n)$ 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 $\mathbb {S}^n$ into $\mathbb {H}^n$. 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 $n$th Lipschitz homotopy group of $\mathbb {H}^n$. We initiate a study of Lipschitz homotopy groups for sub-Riemannian spaces.