Christopher Nerz: Geometric characterization of asymptotic flatness by the existence of a unique CMC-foliation

In mathematical general relativity, one often considers 'isolated' gravitational systems. This means system such that (almost) the whole mass of each time-slice {t = const.} is contained within a bounded domain. These time-slices are mathematically modeled by asymptotically flat Riemannian manifolds which again are defined by the existence of a coordinate system satisfying suitable decay assumptions. In particular, this definition depends on a choice of coordinates. In this talk, we characterize asymptotic flatness by existence of a suitable foliation by hypersurfaces of constant mean curvature, first constructed by Huisken-Yau. Thus, we conclude that asymptotic flatness is a geometric property, i.e. it is coordinate independent, as to be expected by the physical motivation. Furthermore, we give a new coordinate-free definition of linear momentum which serves as an example how to use this characterization of asymptotic flatness to define physical quantities without using coordinates.