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- W4289790872 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 $1le p<infty$ 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 $nle p<n+1$. 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." @default.
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- W4289790872 date "2011-09-21" @default.
- W4289790872 modified "2023-10-16" @default.
- W4289790872 title "On the lack of density of Lipschitz mappings in Sobolev spaces with Heisenberg target" @default.
- W4289790872 doi "https://doi.org/10.48550/arxiv.1109.4641" @default.
- W4289790872 hasPublicationYear "2011" @default.
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