On complete space-like stationary surfaces in 4-dimensional Minkowski space with graphical Gauss image

Concerning the value distribution problem for generalized Gauss maps, we not only generalize Fujimoto’s theorem (J Math Soc Jpn 40:235–247, 1988) to complete space-like stationary surfaces in ℝ^3,1 , 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, which is determined by the number of solutions of f(w)=w̅ , showing a sharp contrast to Bernstein type results for minimal surfaces in ℝ^4 . Moreover, we introduce the concept of conjugate similarity on SL(2,ℂ) to classify all degenerate stationary surfaces (i.e. m≤ 1 ), and establish several structure theorems for complete stationary graphs in ℝ^3,1 from the viewpoint of the degeneracy of Gauss maps.