  A general criterion in terms of the Schwarzian derivative is given for global
univalence of the Weierstrass--Enneper lift of a planar harmonic mapping.
Results on distortion and boundary regularity are also deduced. Examples are
given to show that the criterion is sharp. The analysis depends on a
generalized Schwarzian defined for conformal metrics and on a Schwarzian
introduced by Ahlfors for curves. Convexity plays a central role.
