The global Cauchy problem for the Einstein equations with T2 symmetry

Since the Einstein equations form a system of hyperbolic equations, the natural framework to study the set of solutions is provided by the initial value formulation. For sufficiently regular initial data, the local well-posedness theory is well understood. However, globally, the solutions of the Einstein equations can and typically do exhibit very complex dynamics. Apart from the regimes where small data results have been established, such as the stability of the Minkowski or de Sitter spaces, understanding the asymptotic behaviour of the solutions is for the moment mainly restricted to certain symmetry classes. This is the approach considered here: I will be focusing on the long time dynamics of solutions for initial data of arbitrarily large size, but which enjoy particular symmetry properties, reducing the Einstein equations to a system of quasi-linear wave equations in 1+1 dimensions. The talk will start with a general introduction to the Cauchy problem in relativity. I will then present certain global existence results for T2 symmetric data with high regularity. If time permits, I will talk about some work in progress in collaboration with Phillipe G. Lefloch concerning the behaviour of T2 symmetric solutions with low regularity.