Conformality and $Q$-harmonicity in sub-Riemannian manifolds
  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. 
Luca Capogna, Giovanna Citti, Enrico Le Donne, Alessandro Ottazzi
math.AP math.DG math.MG