Formulações da massa ADM e gráficos com bordo não compacto

Abstract

The fi rst part of this work consists of demonstrating that the ADM mass of an asymptotically at variety can be calculated in terms of a asymptotic limit of integrals 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 at varieties with non-compact boundary. 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, Girão and Montalbán [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 original idea, we express scalar curvature as the divergence of a vector eld and use the Aleksandrov-Fenchel inequality to obtain lower limits of the boundary integrals.