Abstract
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 \(\mathbb {R}^{3,1}\), 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 \(f(w)=\bar{w}\), showing a sharp contrast to Bernstein type results for minimal surfaces in \(\mathbb {R}^4\). Moreover, we introduce the concept of conjugate similarity on \(SL(2,\mathbb {C})\) to classify all degenerate stationary surfaces (i.e. \(m\le 1\)), and establish several structure theorems for complete stationary graphs in \(\mathbb {R}^{3,1}\) from the viewpoint of the degeneracy of Gauss maps.
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Ou, L., Cheng, C. & Yang, L. On complete space-like stationary surfaces in 4-dimensional Minkowski space with graphical Gauss image. Math. Z. 306, 16 (2024). https://doi.org/10.1007/s00209-023-03408-1
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DOI: https://doi.org/10.1007/s00209-023-03408-1