Heisenberg quasiregular ellipticity

  • Katrin Fässler

    University of Jyväskylä, Finland
  • Anton Lukyanenko

    George Mason University, Fairfax, USA
  • Jeremy T. Tyson

    University of Illinois, Urbana, USA
Heisenberg quasiregular ellipticity cover

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Following the Euclidean results of Varopoulos and Pankka–Rajala, we provide a necessary topological condition for a sub-Riemannian 3-manifold MM to admit a nonconstant quasiregular mapping from the sub-Riemannian Heisenberg group H\mathbb{H}. As an application, we show that a link complement S3\L\mathbb{S}^3\backslash L has a sub-Riemannian metric admitting such a mapping only if LL is empty, an unknot or Hopf link. In the converse direction, if LL is empty, a specific unknot or Hopf link, we construct a quasiregular mapping from H\mathbb{H} to S3\L\mathbb{S}^3\backslash L.

The main result is obtained by translating a growth condition on π1(M)\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.

Cite this article

Katrin Fässler, Anton Lukyanenko, Jeremy T. Tyson, Heisenberg quasiregular ellipticity. Rev. Mat. Iberoam. 35 (2019), no. 2, pp. 471–520

DOI 10.4171/RMI/1060