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

The Gauss map and total curvature of complete minimal Lagrangian surfaces in the complex two-space