  We refine estimates introduced by Balogh and Bonk, to show that the boundary
extensions of isometries between smooth strongly pseudoconvex domains in $\C^n$
are conformal with respect to the sub-Riemannian metric induced by the Levi
form. As a corollary we obtain an alternative proof of a result of Fefferman on
smooth extensions of biholomorphic mappings between pseudoconvex domains. The
proofs are inspired by Mostow's proof of his rigidity theorem and are based on
the asymptotic hyperbolic character of the Kobayashi or Bergman metrics and on
the Bonk-Schramm hyperbolic fillings.
