On the marginally trapped surfaces in Minkowski space-time 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 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 ${\mathbb{S}}_1^4(1)$ or anti-de Sitter space-time ${\mathbb{H}}_1^4(-1)$ 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 ${\mathbb{S}}_1^4(1)$ or ${\mathbb{H}}_1^4(-1)$ with harmonic Gauss map.

References in corpus (1)

• Trapped surfaces, horizons and exact solutions in higher dimensions