# On the lack of density of Lipschitz mappings in Sobolev spaces with Heisenberg target

## Piotr Hajlasz

### (University of Pittsburgh)

Date: Oct 10, 2011, time: 08:30

Place: Centro De Giorgi, Scuola Normale Superiore

Abstract. I am going to talk about my recent joint paper with
N. DeJarnette, A. Lukyanenko, and J. Tyson.
We study the question: When are Lipschitz mappings dense in the Sobolev space $W^{1,p}(M,H^n)$?
Here $M$ denotes a compact Riemannian manifold with or without boundary, while $H^n$ denotes the $n$th Heisenberg group equipped with a sub-Riemannian metric. We show that Lipschitz maps are dense in $W^{1,p}(M,H^n)$ for all $1\le p<\infty$ if $\dim M \le n$, but that Lipschitz maps are not dense in $W^{1,p}(M,H^n)$ 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 $H^n$. The nondensity assertion is strictly related to the fact that the
$n$th Lipschitz homotopy group of $H^n$ is nontrivial. We initiate a study of Lipschitz homotopy groups for sub-Riemannian spaces.