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On the marginally trapped surfaces in Minkowski space-time with finite type Gauss map

In this paper, we work on the marginally trapped surfaces in the 4-dimensional Minkowski, de Sitter and anti-de Sitter space-times. We obtain the complete classification of the marginally trapped surfaces in the Minkowski space-time with pointwise 1-type Gauss map. Further, we give construction of marginally trapped surfaces with 1-type Gauss map and a given boundary curve. We also state some explicit examples. We also prove that a marginally trapped surface in the de Sitter space-time $\mathbb S^4_1(1)$ or anti-de Sitter space-time $\mathbb H^4_1(-1)$ has pointwise 1-type Gauss map if and only if its mean curvature vector is parallel. Moreover, we obtain that there exists no marginally trapped surface in $\mathbb S^4_1(1)$ or $\mathbb H^4_1(-1)$ with harmonic Gauss map.