Heisenberg quasiregular ellipticity

TL;DR

This paper establishes topological conditions for sub-Riemannian 3-manifolds to admit nonconstant quasiregular maps from the Heisenberg group, linking geometric analysis, topology, and potential theory.

Contribution

It provides a necessary topological criterion for quasiregular mappings from the Heisenberg group to sub-Riemannian 3-manifolds, extending Euclidean results to a sub-Riemannian setting.

Findings

Link complements admit such mappings only if they are empty, unknotted, or Hopf links.

Constructs explicit quasiregular maps for certain link complements.

Connects growth conditions of fundamental groups to solutions of the 4-harmonic equation.

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 $S^3\backslash L$ has a sub-Riemannian metric admitting such a mapping only if $L$ is empty, the 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 $S^3\backslash L$. 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.