On the marginally trapped surfaces in 4-dimensional space-times with finite type Gauss map

Abstract

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 a construction of a marginally trapped surface with 1-type Gauss map with 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.