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

Abstract

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.