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

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

2010-10-20

arxiv.org/abs/1010.4256v1

Mathematics

Differential Geometry

10 pages, 1 figure

Scientific paper

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.

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