Heisenberg quasiregular ellipticity

Abstract

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.