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.
Highlights
Trapped surfaces were introduced by Sir Roger Penrose in [1] and play an important role in cosmology
From a purely differential geometric point of view, a marginally trapped surface in a spacetime is a Riemannian surface whose mean curvature vector field is lightlike at every point, i.e., for every point p of the surface, the mean curvature vector H ( p) satisfies h H ( p), H ( p)i = 0 and
Most of the results give a complete classification of marginally trapped surfaces in a specific spacetime under one or more additional geometric conditions, such as having positive relative nullity [2], having parallel mean curvature vector field [3], having finite type Gauss map [4], being invariant under certain 1-parameter groups of isometries [5,6,7] or being isotropic [8]
Summary
Trapped surfaces were introduced by Sir Roger Penrose in [1] and play an important role in cosmology. Most of the results give a complete classification of marginally trapped surfaces in a specific spacetime under one or more additional geometric conditions, such as having positive relative nullity [2], having parallel mean curvature vector field [3], having finite type Gauss map [4], being invariant under certain 1-parameter groups of isometries [5,6,7] or being isotropic [8] Several of these results and the above mentioned generalizations are due to Bang-Yen Chen and his collaborators and we should mention his 2009 overview paper on the topic [9]. We end the paper with some open problems (Section 9)
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