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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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On the lack of density of Lipschitz mappings in Sobolev spaces with Heisenberg target
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by Noel DeJarnette, Piotr Hajłasz, Anton Lukyanenko and Jeremy T. Tyson PDF
Conform. Geom. Dyn. 18 (2014), 119-156 Request permission

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
  • MR Author ID: 332316
  • 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
  • MR Author ID: 625886
  • Email: tyson@math.uiuc.edu
  • 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”.
  • © Copyright 2014 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 18 (2014), 119-156
  • MSC (2010): Primary 46E35, 30L99; Secondary 46E40, 26B30, 53C17, 55Q40, 55Q70
  • DOI: https://doi.org/10.1090/S1088-4173-2014-00267-7
  • MathSciNet review: 3226622