Subvariedades atrapadas en espaciotiempos lorentzianos

Resumo

In this memory, our research is developed in the case of codimension two spacelike submanifolds which are immersed in a spacetime M of dimension n+2. That is, we consider a Lorentzian manifold M of dimension n+2 timelike oriented and a codimension two spacelike submanifold N immersed in M. In this context we set different objetives. On the one hand, our main aim is to study trapped submanifolds. If we denote by H the mean curvature vector field of the submanifold, we say that N is trapped (resp. marginally trapped; weakly trapped) if H is timelike (resp. null; causal). This kind of submanifolds have attracted interest since Penrose introduced them in 1965 in General Relativity to study spacetime singularities and black holes. We have studied trapped submanifolds immersed in different Lorentzian spacetimes: the Lorentz-Minkowski spacetime (Chapter 3), de Sitter spacetime (Chapter 4) and the family of the general Robertson-Walker spacetimes (Chapter 6). On the other hand, we focus on the particular situation in which the codimension two spacelike submanifold, N, is contained in a null hypersurface S of the spacetime M. We say then that the submanifold factorizes through S. Null hypersurfaces have an interesting geometry and they also play an important role in General Relativity, where they arise as black hole event horizons and Cauchy horizons. In this case, we know that if N factorizes through a null hypersurface, then there always exists a globally defined normal null frame which is future-pointing. In our work we see that, when working in the Lorentz-Minkowski spacetime or de Sitter spacetime, this allows us to codify the intrinsic and extrinsic geometries of the submanifold in terms of a unique positive function on N (Chapters 3 and 4). In this memory we also study (Chapter 5) a natural correspondence between the light cone of the Lorentz-Minkowski spacetime and the also called light cones of de Sitter and anti-de Sitter spacetimes. To obtain our results we have used, among other techniques, the weak maximum principle for the Laplacian, applying it for the case of stochastically complete submaniolds. We have also studied the geometry of our submanifolds in terms of the globally defined normal null frame whose existence is assured in the case in which the submanifold factorizes through a null hypersurface. Some of our more relevant results are the following. In Chapter 3 we classify codimension two totally umbilical spacelike submanifolds which fatorize through the light cone of the Lorentz-Minkowski spacetime and we also give non-existence results for those which are weakly trapped. In Chapter 4 we characterize codimension two compact marginally trapped submanifolds factorizing through the light cone of de Sitter spacetime. In Chapter 5 we establish a natural correspondence between the light cone of the Lorentz-Minkowski spacetime and the also called light cones of de Sitter and anti-de Sitter spacetimes. Finally, in Chapter 6 we give some rigidity results for marginally trapped submanifolds, with applications to some cases with physical relevance.