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

Abstract. In this paper, we show how to reduce the Penrose conjecture to the known Riemannian Penrose inequality case whenever certain geometrically motivated systems of equations can be solved. Whether or not these special systems of equations have general existence theories is therefore an important open problem. The key tool in our method is the derivation of a new identity which we call the generalized Schoen-Yau identity, which is of independent interest. Using a generalized Jang equation, we use this identity to propose canonical embeddings of Cauchy data into corresponding static spacetimes. In addition, we discuss the Carrasco-Mars counterexample to the Penrose conjecture for generalized apparent horizons (added since the first version of this paper was posted on the arXiv) and instead conjecture the Penrose inequality for time-independent apparent horizons, which we define.

The Positive Mass Theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass. The Riemannian Penrose inequality provides a sharp lower bound for the mass when black holes are present. More precisely, this lower bound is given in terms of the area of an outermost minimal surface, and equality is achieved only for Schwarzschild metrics. The Riemannian Penrose inequality was first proved in three dimensions in 1997 by G. Huisken and T. Ilmanen for the case of a single black hole [HI01]. In 1999, H. Bray extended this result to the general case of multiple black holes using a different technique [Bra01]. In this paper we extend the technique of [Bra01] to dimensions less than 8.

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We introduce a generalized version of the Jang equation, designed for the general case of the Penrose Inequality in the setting of an asymptotically flat space-like hypersurface of a spacetime satisfying the dominat energy condition. The appropriate existence and regularity results are established in the special case of spherically symmetric Cauchy data, and are applied to give a new proof of the general Penrose Inequality for these data sets. When appropriately coupled with an inverse mean curvature flow, analogous existence and regularity results for the associated system of equations in the nonspherical setting would yield a proof of the full Penrose Conjecture. Thus it remains as an important and challenging open problem to determine whether this system does indeed admit the desired solutions.