Abstract
In general relativity, the ADM mass is a notion of total mass of an isolated system, modeled by asymptotically flat manifolds. These manifolds describe gravitating systems in which all the matter is contained inside a compact region. In the first part of this dissertation, we generalize the formulas for the ADM mass derived by Miao (2020) and demonstrate that the mass of an n-dimensional asymptotically flat manifold can be understood as the limit of the total mean curvature plus the total defect of the dihedral angle of the boundary of large Riemannian polyhedra. Additionally, we express the n-dimensional mass as a suitable integral of geometric quantities that determine the (n-1)-dimensional mass of a coordinate hyperplane.
In the second part, we investigate the Bartnik quasi-local mass, which is a notion introduced by Bartnik (1989) to describe the mass contained in a bounded region. We consider data with nonnegative Gauss curvature and positive mean curvature for which we present and discuss some estimates on the Bartnik mass in terms of the area, the total mean curvature, and a quantity measuring the roundness of the metric. Notably, if the ratio between the maximum and the minimum of the Gauss curvature approaches 1, we show that our main estimate converges to the sharp bound derived by Miao (2009) for round spheres with positive mean curvature. Furthermore, if the total mean curvature approaches 0, our estimate tends towards the sharp bound found by Mantoulidis and Schoen (2015) for the black hole horizon case. We then discuss some consequences of our argument in the setting of nonnegative scalar curvature fill-ins of a closed manifold with positive scalar curvature.