Proof of the Riemannian Penrose Inequality Using the Positive Mass Theorem

Abstract

We prove the Riemannian Penrose Conjecture, an important case of a conjecture [41] made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of asymptotically flat Riemannian 3-manifolds with nonnegative scalar curvature which contain minimal spheres. In particular, if we consider a Riemannian 3-manifold as a totally geodesic submanifold of a space-time in the context of general relativity, then outermost minimal spheres with total area A correspond to apparent horizons of black holes contributing a mass A/16π, scalar curvature corresponds to local energy density at each point, and the rate at which the metric becomes flat at infinity corresponds to total mass (also called the ADM mass). The Riemannian Penrose Conjecture then states that the total mass of an asymptotically flat 3-manifold with nonnegative scalar curvature is greater than or equal to the mass contributed by the black holes.The flow of metrics we define continuously evolves the original 3-metric to a Schwarzschild 3-metric, which represents a spherically symmetric black hole in vacuum. We define the flow such that the area of the minimal spheres (which flow outward) and hence the mass contributed by the black holes in each of the metrics in the flow is constant, and then use the Positive Mass Theorem to show that the total mass of the metrics is nonincreasing. Then since the total mass equals the mass of the black hole in a Schwarzschild metric, the Riemannian Penrose Conjecture follows.We also refer the reader to the beautiful work of Huisken and Ilmanen [30], who used inverse mean curvature flows of surfaces to prove that the total mass is at least the mass contributed by the largest black hole.In Sections 1 and 2, we motivate the problem, discuss important quantities like total mass and horizons of black holes, and state the Positive Mass Theorem and the Penrose Conjecture for Riemannian 3-manifolds. In Section 3, we give the proof of the Riemannian Penrose

Positivity of the universal pairing in 3 dimensions

“…Since S is minimal in AB, it has vanishing mean curvature. This implies, by arguments of Miao [31] or Bray [5], arising from work on the Riemannian Penrose inequality, that the metrics on AA and BB can be uniformly approximated by C ∞ Riemannian metrics satisfying uniform pointwise scalar curvature bounds R ≥ −6. Technology due to Miles Simon [43] lets one apply short-time Ricci flow to such singular metrics, which preserves the pointwise scalar curvature bounds.…”

“…The singular metric on AA k (away from the orbifold locus) is of this kind, since it is obtained by doubling a genuine Riemannian metric (also see Bray [5], equation 102 and the surrounding text for an explicit estimate). After we have obtained such an approximating metric, observe since AA k is compact, that there is some uniform pointwise bound on the curvature and all its covariant derivatives.…”

“…5 So to complete the proof, assume that (A, S 0 ) is nonorientable, RP 2 -irreducible, and boundary irreducible, and that ∂A = S 0 is a (possibly noncompact) surface of finite type other than a disk or Möbius band. It suffices to show that there is no distinct (A , S 0 ) with identical orientation covers A ∼ = A (rel S 0 ).…”

“…Since our surfaces S k , k ≤ ∞, have no uniform size tubular neighborhoods to work with, we follow Bray [5] and desingularize by adding an untapered "plug" of thickness 2δ over S k . Note that the constructions both of Bray and of Miao commute with isometries, so that in practice we perform the desingularization in the cover and then define AA δ k to be the quotient orbifold.…”

A note on asymptotically flat metrics on ℝ³ which are scalar-flat and admit minimal spheres

“…A sharpening of the Positive Mass Theorem known as the Penrose Inequality has recently been attained by Bray [Br1] and Huisken-Ilmanen [HI] independently. We state the version from [Br1]. We call a minimal sphere outermost with respect to an end if it is not enclosed by any minimal surface in the exterior region containing this end (cf.…”