Structure of singularities of spacetimes with toroidal or hyperbolic symmetry

I will present recent results concerning the study of the global Cauchy problem in general relativity under symmetry assumptions.

More specifically, I will be focusing on the structure of singularities and the uniqueness in the large for solutions of the Einstein equations, the so-called strong cosmic censorship, under the assumption that the initial data is given on some compact manifold with prescribed symmetry.

In particular, I will present some results which concerned the asymptotic behaviour of the area of the orbits of symmetry, a quantity which plays in important role for the study of these solutions. From the point of view of PDE, this corresponds to a global existence theorem for a system of non-linear 1+1 wave equations.