A geometric study of marginally trapped surfaces in space forms and Robertson-Walker spacetimes -- an overview

TL;DR

This paper reviews the differential geometry of marginally trapped surfaces across various spacetimes, summarizing classifications, properties, and examples, and highlighting open questions in the field.

Contribution

It provides a comprehensive overview of the local descriptions, classifications, and examples of marginally trapped surfaces in different spacetime models.

Findings

Classified marginally trapped surfaces with special geometric properties

Presented examples of constant Gaussian curvature marginally trapped surfaces

Outlined open problems in the geometric study of these surfaces

Abstract

A marginally trapped surface in a spacetime is a Riemannian surface whose mean curvature vector is lightlike at every point. In this paper we give an up-to-date overview of the differential geometric study of these surfaces in Minkowski, de Sitter, anti-de Sitter and Robertson-Walker spacetimes. We give the general local descriptions proven by Anciaux and his coworkers as well as the known classifications of marginally trapped surfaces satisfying one of the following additional geometric conditions: having positive relative nullity, having parallel mean curvature vector field, having finite type Gauss map, being invariant under a one-parameter group of ambient isometries, being isotropic, being pseudo-umbilical. Finally, we provide examples of constant Gaussian curvature marginally trapped surfaces and state some open questions.