  We derive a quasiconformal extension to 3-space of the Weierstrass-Enneper
lifts of a class of harmonic mappings defined in the unit disk. The extension
is based on fibrations of space by circles in domain and image that correspond
to each other in a natural way. Convexity plays an essential role in the
analysis. As a corollary we derive a sufficient condition for the underlying
harmonic mapping to be univalent in the disk, with an explicit quasiconformal
extension to the extended plane that generalizes the well known formula by
Ahlfors-Weill.
