Conformality and $Q$-harmonicity in sub-Riemannian manifolds

TL;DR

This paper establishes the equivalence of various conformal map notions in sub-Riemannian manifolds, proving smoothness of 1-quasiconformal maps under certain regularity conditions, especially in contact manifolds.

Contribution

It introduces a sub-Riemannian framework for p-harmonic coordinates and demonstrates regularity propagation, advancing understanding of conformal maps in these geometries.

Findings

Proves equivalence of conformal map notions in sub-Riemannian manifolds.

Shows 1-quasiconformal maps are smooth in manifolds with regular subelliptic p-Laplacian theory.

Establishes that contact manifolds support the necessary regularity.

Abstract

We prove the equivalence of several natural notions of conformal maps between sub-Riemannian manifolds. Our main contribution is in the setting of those manifolds that support a suitable regularity theory for subelliptic $p$-Laplacian operators. For such manifolds we prove a Liouville-type theorem, i.e., 1-quasiconformal maps are smooth. In particular, we prove that contact manifolds support the suitable regularity. The main new technical tools are a sub-Riemannian version of p-harmonic coordinates and a technique of propagation of regularity from horizontal layers.