The objective of this talk is to give a new point of view for the validity of Fefferman's mapping theorem from 1974. This result states that a biholomorphism between two smoothly bounded strictly pseudoconvex domains in C^n extends as a smooth diffeomorphism between their closures.
Following ideas from Gromov, Mostow, and Pansu, we discuss a method of proof in the context of quasi-conformal geometry. In particular, we show that every isometry between smoothly bounded strictly pseudoconvex domains is 1-quasi-conformal with respect to the sub-Riemannian distance defined by the Levi form on the boundaries. Subsequently, a PDE argument shows that such maps are smooth. This method was proposed by M. Cowling, and it has been implemented in collaboration with L. Capogna, G. Citti, and A. Ottazzi.