The asymptotically flat manifolds considered in the article are 3-dimensional Riemannian manifolds whose geometry approaches the Euclidean geometry at infinity. Provided a suitable decay rate is fulfilled, they admit global asymptotic invariants, most importantly the ADM mass and the center of mass, originally defined from the Hamiltonian formulation of general relativity.
In recent years, a large number of different techniques have been introduced to define or to recover these quantities.
The authors of the present work investigate a technique based on the isoperimetric deficit of surfaces that enclose large volumes in the asymptotic region. This notion was introduced by G. Huisken in a contribution contained in [P. Chruściel (ed.) et al., Oberwolfach Rep. 3, No. 1, 87–88 (2006; Zbl 1109.83300)]. Here G. Huisken proved that this deficit converges to the ADM mass as the radii of the coordinate spheres go to infinity.
The authors of the present paper first recover the same limit using other isoperimetric deficits that involve the enclosed volume, the area and the total mean curvature of the surfaces. They also get the existence of a stable foliation of the asymptotic region by surfaces that satisfy some curvature conditions (involving the mean and the Gaussian curvatures).
Then, they extend these results to the class of asymptotically flat 3-manifolds with a non-compact boundary, for which the Euclidean half-space is the model space. In this setting, large spheres are replaced by large hemispheres and the various definitions (mass, center of mass, isoperimetric deficits) are revisited accordingly. The foliation constructed here is made of stable isoperimetric surfaces, thus extending the corresponding result obtained by M. Eichmair and J. Metzger [Invent. Math. 194, No. 3, 591–630 (2013; Zbl 1297.49078)] for asymptotically flat manifolds.