A Liouville type theorem for Carnot groups

TL;DR

This paper proves a Liouville type theorem for all Carnot groups by leveraging the smoothness of 1-quasiconformal maps and Tanaka prolongation theory, extending previous results limited to specific groups.

Contribution

It demonstrates that a Liouville type theorem holds universally for Carnot groups using Tanaka prolongation, building on the smoothness of 1-quasiconformal maps.

Findings

Liouville type theorem applies to all Carnot groups

1-quasiconformal maps are smooth in Carnot groups

Tanaka prolongation is key to the proof

Abstract

L. Capogna and M. Cowling showed that if $\phi$ is 1-quasiconformal on an open subset of a Carnot group G, then composition with $\phi$ preserves Q-harmonic functions, where Q denotes the homogeneous dimension of G. Then they combine this with a regularity theorem for Q-harmonic functions to show that $\phi$ is in fact $C^\infty$. As an application, they observe that a Liouville type theorem holds for some Carnot groups of step 2. In this article we argue, using the Engel group as an example, that a Liouville type theorem can be proved for every Carnot group. Indeed, the fact that 1-quasiconformal maps are smooth allows us to obtain a Liouville type theorem by applying the Tanaka prolongation theory.