Penrose-type inequalities with a Euclidean background

Details Penrose-type inequalities with a Euclidean background Penrose-type inequalities with a Euclidean background

2011-08-19

arxiv.org/abs/1108.4042v1

Mathematics

Differential Geometry

15 pages

Scientific paper

We recall the Riemannian Penrose inequality, a geometric statement that restricts the asymptotics of manifolds of nonnegative scalar curvature that contain compact minimal hypersurfaces. The inequality is known for manifolds of dimension up to seven. One source of motivation for the present work is to prove a weakened inequality in all dimensions for conformally flat manifolds. Along the way, we establish new inequalities, including some that apply to manifolds that are merely conformal to a particular type of scalar flat manifold (not necessarily conformally flat). We also apply the techniques to asymptotically flat manifolds containing zero area singularities, objects that generalize the naked singularity of the negative mass Schwarzschild metric. In particular we derive a lower bound for the ADM mass of a conformally flat, asymptotically flat manifold containing any number of zero area singularities.

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