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 $W^{1,p}(M,{\mathbf{H}}^n)$? 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.

Cited by in corpus (1)

• Lipschitz homotopy groups of the Heisenberg groups