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 subRiemannian 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 subRiemannian spaces.
 Publication:

arXiv eprints
 Pub Date:
 September 2011
 arXiv:
 arXiv:1109.4641
 Bibcode:
 2011arXiv1109.4641D
 Keywords:

 Mathematics  Functional Analysis;
 Primary: 46E35;
 30L99;
 Secondary: 46E40;
 26B30;
 53C17;
 55Q40;
 55Q70