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

TL;DR

This paper explores the relationship between total curvature and the Gauss map of complete minimal Lagrangian surfaces in complex two-space, establishing bounds on exceptional values and value distribution properties.

Contribution

It determines the maximum number of exceptional values of the Gauss map and shows that omitting three values implies infinite recurrence of other values for such surfaces.

Findings

Maximum number of exceptional values of the Gauss map is precisely determined.

If the Gauss map omits three values, it takes all other values infinitely often.

Provides new insights into the value distribution of the Gauss map for minimal Lagrangian surfaces.

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.