  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. Later, their decay assumptions were weakened by
Metzger, Huang, Eichmair-Metzger, and the author. 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, and 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 give 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.
