In the last ten years, certain geometric flows have been discovered to have important applications to general relativity. In particular, both the Huisken-Ilmanen proof (for one black hole) and the Bray proof of the Riemannian Penrose Inequality use geometric flows, even if the two approaches are quite different. The Riemannian Penrose Inequality is a statement about time-symmetric (zero second fundamental form), 3 dimensional, space-like slices $(M^3,g)$ of a 3+1 dimensional space-time which is asymptotically flat (meaning that the space-time is asymptotic to the Minkowski space-time at infinity in a certain sense). The assumption of nonnegative energy density in the space-time implies that $(M^3,g)$ has nonnegative scalar curvature, and the total mass of the space-time corresponds to the rate at which $(M^3,g)$ becomes flat at infinity. Furthermore, outermost embedded minimal spheres in $(M^3,g)$ (spheres which are not enclosed by other embedded minimal spheres) correspond to the apparent horizons of black holes, contributing a combined mass equal to $\sqrt{A/16\pi}$. Thus, the natural physical conjecture that the total mass of $(M^3,g)$ is at least the combined mass contributed by the black holes implies a geometric statement bounding the areas of the outermost minimal spheres of $(M^3,g)$ in terms of the asymptotic behavior of $(M^3,g)$ at infinity, assuming $(M^3,g)$ has nonnegative scalar curvature. This geometric statement is known as the Riemannian Penrose Inequality. In this talk, we will discuss the two geometric flow proofs of this inequality as well as recent generalizations and applications of these same methods to general relativity and to computing the Yamabe constants of 3-manifolds.
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