Remarks on the Gauss images of complete minimal surfaces in Euclidean four-space

TL;DR

This paper systematically studies the Gauss map images of complete minimal surfaces in Euclidean four-space, providing geometric interpretations and optimal bounds for exceptional values across various classes of minimal surfaces.

Contribution

It offers new geometric insights and optimal bounds on the number of exceptional values of the Gauss map for different types of minimal surfaces in Euclidean four-space.

Findings

Maximal number of exceptional values for orientable minimal surfaces determined.

Optimal bounds established for minimal Lagrangian surfaces in complex two-space.

Results extended to nonorientable minimal surfaces in Euclidean four-space.

Abstract

We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number of exceptional values of the Gauss map of a complete orientable minimal surface in Euclidean four-space. We also provide optimal results for the maximal number of exceptional values of the Gauss map of a complete minimal Lagrangian surface in the complex two-space and the generalized Gauss map of a complete nonorientable minimal surface in Euclidean four-space.