Asymptotic analysis of isolated gravitational systems

A central result within mathematical relativity is the well-posedness of the initial value formulation of GR for initial data satisfying the Einstein constraint equations. In this context, isolated gravitational systems are modelled by asymptotically Eu- clidean initial data sets and there are physically reasonable expectations about the asymptotic behaviour of these systems. It turns out that a rigorous mathemat- ical understanding of the conditions that guarantee these expectations is still an open problem. This is crucial for canonically conserved quantities carrying physical information to be well-defined. In the case of the ADM energy and linear momen- tum, precise geometric criteria making them well-defined are well-known, but for the ADM centre of mass and angular momentum this is not the case and ad-hoc asymptotic conditions tend to be demanded. In this talk, we will report on some recent results on regularity problems and asymptotic analysis of geometric partial differential equations which allow one to partially characterise, in pure geometric terms, those initial data sets which indeed obey these expected asymptotic proper- ties. In particular, we shall address a Riemannian version of a conjecture posed by Cederbaum-Sakovich concerning constant mean curvature asymptotic foliations in AE manifolds and establish geometric criteria guaranteeing the convergence of the ADM centre of mass.