Heisenberg quasiregular ellipticity

Katrin Fässler, Anton Lukyanenko, Jeremy T. Tyson

Research output: Contribution to journalArticlepeer-review


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 H. As an application, we show that a link complement S3\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 H to S3\L. The main result is obtained by translating a growth condition on π1(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.

Original languageEnglish (US)
Pages (from-to)471-520
Number of pages50
JournalRevista Matematica Iberoamericana
Issue number2
StatePublished - 2019


  • 3-sphere
  • Capacity
  • Contact manifold
  • Hopf link
  • Isoperimetric inequality
  • Link complement
  • Morphism property
  • Nonlinear potential theory
  • Quasiregular mapping
  • Sobolev–Poincaré inequality
  • Sub-Riemannian manifold

ASJC Scopus subject areas

  • Mathematics(all)


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