O teorema da massa positiva e a desigualdade de Penrose para gráficos

Abstract

In this paper we will work with complete Riemannian manifolds, asymptotically flat, that are smooth graphics over Rn. We present an elegant and direct proof for the Positive Mass Theorem. Expressing its scalar curvature as a divergent feld, we will show that the ADM mass of the manifold can be expressed as an integral over the manifold of the product of scalar curvature and a nonnegative potential function. As an application, we will prove also the Penrose inequality giving a lower bound for the boundary integral using the Aleksandrov-Fenchel inequality.