The Riemannian Penrose Inequality with Matter Density

Riemannian Penrose Inequalities are precise geometric statements that imply that the total mass of a zero second fundamental form slice of a spacetime is at least the mass contributed by the black holes, assuming that the spacetime has nonnegative matter density everywhere. In this paper, we remove this last assumption, and prove stronger statements that the total mass is at least the mass contributed by the black holes, plus a contribution coming from the matter density along the slice. We use the first author's conformal flow to achieve this, combined with Stern's harmonic level set techniques in the first case, and spinors in the second case. We then compare these new results to results previously known from Huisken-Ilmanen's inverse mean curvature flow techniques.