On the Geometry of the Level Sets of Bounded Static Potentials

In this paper we present a new approach to the study of asymptotically flat static metrics arising in general relativity. In the case where the static potential is bounded, we introduce new quantities which are proven to be monotone along the level set flow of the potential function. We then show how to use these properties to detect the rotational symmetry of the static solutions, deriving a number of sharp inequalities. Among these, we prove the validity—without any dimensional restriction—of the Riemannian Penrose Inequality, as well as of a reversed version of it, in the class of asymptotically flat static metrics with connected horizon. As a consequence of our analysis, a simple proof of the classical 3-dimensional Black Hole Uniqueness Theorem is recovered and some geometric conditions are discussed under which the same statement holds in higher dimensions.