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

TL;DR

This paper classifies marginally trapped surfaces with pointwise 1-type Gauss maps in Minkowski space-time, constructs examples, and explores conditions in de Sitter and anti-de Sitter spaces, revealing links to mean curvature vector properties.

Contribution

It provides a complete classification of such surfaces in Minkowski space and characterizes their Gauss map types in de Sitter and anti-de Sitter spaces.

Findings

Complete classification of marginally trapped surfaces with pointwise 1-type Gauss map in Minkowski space.

Construction methods for marginally trapped surfaces with prescribed boundary curves.

Proof that surfaces in de Sitter and anti-de Sitter spaces have pointwise 1-type Gauss map iff their mean curvature vector is parallel.

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 $\mathbb S^4_1(1)$ 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.