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Conformal Geometry and Dynamics
Conformal Geometry and Dynamics
ISSN 1088-4173

 

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


Authors: Noel DeJarnette, Piotr Hajłasz, Anton Lukyanenko and Jeremy T. Tyson
Journal: Conform. Geom. Dyn. 18 (2014), 119-156
MSC (2010): Primary 46E35, 30L99; Secondary 46E40, 26B30, 53C17, 55Q40, 55Q70
Published electronically: July 1, 2014
MathSciNet review: 3226622
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Abstract: We study the question: When are Lipschitz mappings dense in the Sobolev space $ W^{1,p}(M,\mathbb{H}^n)$? Here $ M$ denotes a compact Riemannian manifold with or without boundary, while $ \mathbb{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,\mathbb{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,\mathbb{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 $ \mathbb{S}^n$ into $ \mathbb{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 $ \mathbb{H}^n$. We initiate a study of Lipschitz homotopy groups for sub-Riemannian spaces.


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Additional Information

Noel DeJarnette
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green St., Urbana, Illinois 61801
Email: ndejarne@illinois.edu

Piotr Hajłasz
Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260
Email: hajlasz@pitt.edu

Anton Lukyanenko
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green St., Urbana, Illinois 61801
Email: lukyane2@illinois.edu

Jeremy T. Tyson
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green St., Urbana, Illinois 61801
Email: tyson@math.uiuc.edu

DOI: http://dx.doi.org/10.1090/S1088-4173-2014-00267-7
PII: S 1088-4173(2014)00267-7
Received by editor(s): January 29, 2014
Received by editor(s) in revised form: April 3, 2014
Published electronically: July 1, 2014
Additional Notes: The first author acknowledges support from NSF grants DMS 0838434 “EMSW21-MCTP: Research Experiences for Graduate Students”, DMS 0901620 and DMS 0900871. The first author also acknowledges the Department of Mathematics at the University of Pittsburgh for its hospitality during the academic year 2009–2010.
The second author acknowledges support from NSF grant DMS 0900871 “Geometry and topology of weakly differentiable mappings into Euclidean spaces, manifolds and metric spaces”.
The third author acknowledges support from NSF grant DMS 0838434 “EMSW21-MCTP: Research Experiences for Graduate Students”.
The fourth author acknowledges support from NSF grants DMS 0555869 “Nonsmooth methods in geometric function theory and geometric measure theory on the Heisenberg group” and DMS 0901620 “Geometric analysis in Carnot groups”.
Article copyright: © Copyright 2014 American Mathematical Society