Résume  The concept of trapped surfaces was originally formulated by Penrose for the case of 2dimensional spacelike surfaces in 4dimensional spacetimes in terms of the signs or the vanishing of the socalled null expansions. This is obviously related to the causal orientation of the mean curvature vector of the surface, which provides a better and powerful characterization of the trapped surfaces and allows the generalization of this concept to codimension two spacelike submanifolds of arbitrary dimension n.
In this sense, an ndimensional spacelike submanifold Σ of an (n + 2) dimensional spacetime is said to be future trapped if its mean curvature vector field H is timelike and futurepointing everywhere on Σ, and similarly for past trapped. If H is lightlike (or null) and futurepointing everywhere on Σ then the submanifold is said to be marginally future trapped, and similarly for marginally past trapped.
Finally, if H is causal and futurepointing everywhere, the submanifold is said to be weakly future trapped, and similarly for weakly past trapped. The extreme case H = 0 on Σ corresponds to a minimal submanifold.
In this lecture we consider codimension two compact marginally trapped sub manifolds in the light cone of de Sitter space. In particular, we show that they are conformally diffeomorphic to the round sphere and, as an application of the solution of the Yamabe problem on the round sphere, we derive a classification result for such submanifolds. We also fully describe the codimension two compact marginally trapped submanifolds contained into the past infinite of the steady state space.
This is part of our work in progress with Verónica L. Cánovas (from Murcia) and Marco Rigoli (from Milano).
