Chapter 9 - On the Riemannian Penrose inequality

Summary

In 1973 Roger Penrose proposed an inequality which states that, in an isolated gravitational system with nonnegative local energy density, the total mass of the system must be at least as much as that contributed by any black holes contained within. The original motivation for proposing the inequality is the fact that a counterexample of this inequality would produce a counterexample for the cosmic censorship conjecture. While the Penrose inequality in full generality remains an open problem, we present in this chapter the main ideas of three proofs of the Riemannian Penrose inequality, a special case that arises in the time-symmetric setting. In this setting, the Penrose inequality is formulated on an asymptotically flat manifold with nonnegative scalar curvature that contains closed minimal surfaces, and asserts a bound of the total mass in terms of the area of the outermost minimal surface. We begin with Lam's proof for the graphical case, then move to the inverse mean curvature flow approach of Huisken and Ilmanen, and finish with a sketch of Bray's conformal flow technique.