A massa e o centro de massa de gráficos.

Abstract

In this thesis, masses of asymptotically flat and asymptotically hyperbolic graphs are computed as volume integrals on the manifolds. This is done by using the mass definition via Einstein’s tensor and applying the divergence theorem. A brief digression is made and an expression for the center of mass of an asymptotically flat graph is presented. The positive mass and Penrose inequalities are considered in this thesis. The validity of these inequalities are proved for the special cases we are considering. For the proof of the Penrose inequality, the boundary term of the integral that expresses the mass is computed and known results, like the Alexandrov-Fenchel inequality, are used. This thesis is supported on the works of Lam, for codimension one asymptotically flat graphs, of Mirandola-Vit´orio, for arbitrary codimension asymptotically flat graphs, and the works of Dahl-Gicquaud-Sakovich and de Lima-Gir˜ao for codimension one hyperbolic graphs.