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Revista Matemática Iberoamericana

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Volume 35, Issue 2, 2019, pp. 471–520
DOI: 10.4171/rmi/1060

Published online: 2019-02-05

Heisenberg quasiregular ellipticity

Katrin Fässler[1], Anton Lukyanenko[2] and Jeremy T. Tyson[3]

(1) University of Jyväskylä, Finland
(2) George Mason University, Fairfax, USA
(3) University of Illinois, Urbana, USA

Following the Euclidean results of Varopoulos and Pankka–Rajala, we provide a necessary topological condition for a sub-Riemannian 3-manifold $M$ to admit a nonconstant quasiregular mapping from the sub-Riemannian Heisenberg group $\mathbb{H}$. As an application, we show that a link complement $\mathbb{S}^3\backslash L$ has a sub-Riemannian metric admitting such a mapping only if $L$ is empty, an unknot or Hopf link. In the converse direction, if $L$ is empty, a specific unknot or Hopf link, we construct a quasiregular mapping from $\mathbb{H}$ to $\mathbb{S}^3\backslash L$.

The main result is obtained by translating a growth condition on $\pi_1(M)$ into the existence of a supersolution to the 4-harmonic equation, and relies on recent advances in the study of analysis and potential theory on metric spaces.

Keywords: Quasiregular mapping, contact manifold, sub-Riemannian manifold, 3-sphere, link complement, Hopf link, isoperimetric inequality, Sobolev–Poincar´e inequality, capacity, nonlinear potential theory, morphism property

Fässler Katrin, Lukyanenko Anton, Tyson Jeremy: Heisenberg quasiregular ellipticity. Rev. Mat. Iberoam. 35 (2019), 471-520. doi: 10.4171/rmi/1060