  The purpose of this paper is to reveal the relationship between the total
curvature and the global behavior of the Gauss map of a complete minimal
Lagrangian surface in the complex two-space. To achieve this purpose, we show
the precise maximal number of exceptional values of the Gauss map for a
complete minimal Lagrangian surface with finite total curvature in the complex
two-space. Moreover, we prove that if the Gauss map of a complete minimal
Lagrangian surface which is not a Lagrangian plane omits three values, then it
takes all other values infinitely many times.
