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We revisit the interplay between the mass, the center of mass and the largescale behavior of certain isoperimetric quotients in the setting ofasymptotically flat $3$-manifolds (both without and with a non-compactboundary). In the boundaryless case, we first check that the isoperimetricdeficits involving the total mean curvature recover the ADM mass in theasymptotic limit, thus extending a classical result due to G. Huisken. Next,under a Schwarzschild asymptotics and assuming that the mass is positive weindicate how the implicit function method pioneered by R. Ye and refined byL.-H. Huang may be adapted to establish the existence of a foliation of aneighborhood of infinity satisfying the corresponding curvature conditions.Recovering the mass as the asymptotic limit of the corresponding relativeisoperimetric deficit also holds true in the presence of a non-compactboundary, where we additionally obtain, again under a Schwarzschildasymptotics, a foliation at infinity by free boundary constant mean curvaturehemispheres, which are shown to be the unique relative isoperimetric surfacesfor all sufficiently large enclosed volume, thus extending to this setting acelebrated result by M. Eichmair and J. Metzger. Also, in each case treatedhere we relate the geometric center of the foliation to the center of mass ofthe manifold as defined by Hamiltonian methods.

Mass, center of mass and isoperimetry in asymptotically flat $3$-manifolds