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

Piotr Hajlasz

(University of Pittsburgh)

Event: ERC Workshop on Geometric Analysis on sub-Riemannian and Metric Spaces

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.