Geometric characterizations of asymptotic flatness and linear momentum in general relativity

In 1996, Huisken–Yau proved that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed surfaces of constant mean curvature (CMC) if it is asymptotically equal to the (spatial) Schwarzschild solution. Their decay assumptions were weakened by Metzger, Huang, Eichmair–Metzger, and the author at a later stage. In this work, we prove the reverse implication, i.e. any three-dimensional Riemannian manifold is asymptotically flat if it possesses a CMC-cover satisfying certain geometric curvature estimates, a uniqueness property, a weak foliation property, while each surface has weakly controlled instability. With the author's previous result that every asymptotically flat manifold possesses a CMC-foliation, we conclude that asymptotic flatness is characterized by existence of such a CMC-cover. Additionally, we use this characterization to provide a geometric (i.e. coordinate-free) definition of a (CMC-)linear momentum and prove its compatibility with the linear momentum defined by Arnowitt–Deser–Misner.