On complete space-like stationary surfaces in Minkowski spacetime with graphical Gauss image

TL;DR

This paper extends value distribution theory for Gauss maps of complete space-like stationary surfaces in Minkowski spacetime, providing bounds on exceptional values, classifying degenerate cases, and contrasting with Euclidean minimal surfaces.

Contribution

It generalizes Fujimoto's theorem to Minkowski spacetime, estimates exceptional values for Gauss images, and introduces conjugate similarity to classify degenerate stationary surfaces.

Findings

Established upper bounds for exceptional values of Gauss maps.

Classified degenerate stationary surfaces using conjugate similarity.

Provided structure theorems for complete stationary graphs in Minkowski spacetime.

Abstract

Concerning the value distribution problem for generalized Gauss maps, we not only generalize Fujimoto's theorem to complete space-like stationary surfaces in Minkowski spacetime, but also estimate the upper bound of the number of exceptional values when the Gauss image lies in the graph of a rational function f of degree m, showing a sharp contrast to Bernstein type results for minimal surfaces in 4-dimensional Euclidean space. Moreover, we introduce the conception of conjugate similarity on the special linear group to classify all degenerate stationary surfaces (i.e. m=0 or 1), and establish several structure theorems for complete stationary graphs in Minkowski spacetime from the viewpoint of the degeneracy of Gauss maps.