Proof of the Riemannian Penrose inequality using the positive mass theorem. (Q1609841): Difference between revisions

An asymptotically flat 3-manifold is a Riemannian manifold \((M^3, g)\) which, outside a compact set, is a disjoint union of one ore more regions (called ends) diffeomorphic to \(({\mathbb R}^3\setminus B_1(0), \delta)\), where the metric \(g\) in each of \({\mathbb R}^3\) coordinate charts approaches the standart metric \(\delta\) on \({\mathbb R}^3\) at infinity. The positive mass theorem and the Penrose conjecture are both statements which refer to a particular chosen end of \((M^3, g)\). The total mass of \((M^3, g)\) is a parameter related to how fast this chosen end of \((M^3, g)\) becomes flat at infinity. The main result of the paper is the proof of the following geometric statement -- the Riemannian Penrose conjecture: Let \((M^3, g)\) be a complete, smooth, asymptotically flat 3-manifold with nonnegative scalar curvature and total mass \(m\) whose outermost minimal spheres have total surface area \(A\). Then \(m\geq\sqrt{\frac{A}{16\pi}}\) with equality if and only if \((M^3, g)\) is isometric to the Schwarzschild metric \(({\mathbb R}^3\setminus\{0\}, s)\) of mass \(m\) outside their respective horizons.