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

TL;DR

This paper explores the relationship between mass, center of mass, and isoperimetric properties in asymptotically flat 3-manifolds, extending classical results and establishing new foliations and uniqueness results in both boundaryless and boundary cases.

Contribution

It extends isoperimetric deficit results to recover ADM mass and constructs foliations near infinity with prescribed curvature conditions in asymptotically flat 3-manifolds.

Findings

Isoperimetric deficits recover ADM mass asymptotically.

Existence of curvature-prescribed foliations near infinity.

Unique large-volume isoperimetric surfaces with boundary conditions.

Abstract

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