Inserted: 12 may 2016
Last Updated: 15 aug 2018
We attach a preliminary and more detailed version, and the final published version, where some proofs were summarized.
We establish regularity of conformal maps between sub-Riemannian manifolds from regularity of $Q$-harmonic functions, and in particular we prove a Liouville-type theorem, i.e., 1-quasiconformal maps are smooth in all contact sub-Riemannian manifolds. Together with some recent results by Capogna and Le Donne, our work yields a new proof of the smoothness of boundary extensions of biholomorphims between strictly pseudoconvex smooth domains.
Keywords: sub-Riemannian geometry, conformal transformation, quasi-conformal maps, subelliptic PDE, harmonic coordinates, Liouville Theorem, Popp measure, morphism property, regularity for p-harmonic functions