Formulações da massa ADM e gráficos com bordo não compacto

Abstract

The first part of this work consists of demonstrating that the ADM mass of an

asymptotically flat variety can be calculated in terms of a asymptotic limit of inte- grals involving the Einstein tensor. For this purpose, we follow the method proposed

by Herzlich ([31]) that relates Michel’s analysis ([51]) for asymptotic invariants and a part-integration formula based on Bianchi’s contracted identity. Given the general character of this approach, we will analize jointly analysing the center of mass and a concept of mass developed for asymptotically hyperbolic manifolds.

In a second moment, we study asymptotically flat varieties with non-compact boun- dary. In this context, we have a similar notion of ADM mass developed by Almaraz,

Barbosa and De Lima ([3]) that allows us to adapt the previous method to express the ADM mass also in terms of geometric tensors. For that, we will follow the article by De Lima, Gir ̃ao and Montalb ́an [24]. Finally, based on the article by Barbosa e Meira [8], we prove a version of Penrose

Inequality for graphic hypersurfaces with non-compact boundary. Following Lam’s ori- ginal idea, we express scalar curvature as the divergence of a vector field and use the

Aleksandrov-Fenchel inequality to obtain lower limits of the boundary integrals.