The Graphs Cases of the Riemannian Positive Mass and Penrose Inequalities in All Dimensions

Abstract

We consider complete asymptotically flat Riemannian manifolds that are the graphs of smooth functions over ${\mathbb{R}}^n$. By recognizing the scalar curvature of such manifolds as a divergence, we express the ADM mass as an integral of the product of the scalar curvature and a nonnegative potential function, thus proving the Riemannian positive mass theorem in this case. If the graph has convex horizons, we also prove the Riemannian Penrose inequality by giving a lower bound to the boundary integrals using the Aleksandrov-Fenchel inequality.

10 pages, 1 figure

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