Mathematics

The isoperimetric mass is an alternative definition due to Huisken of the renowed ADM mass, a quantity ruling the geometry of asymptotically flat 3-manifolds with nonnegative scalar curvature.  In fact, Schoen-Yau's positive mass theorem asserts that in this setting the ADM mass is nonnegative, while Huisken-Ilmanen/Bray' Riemannian Penrose inequality yields a sharp positive lower bound in terms of the area of an outermost minimal boundary.Contrarily to the classical notion of ADM mass, the isoperimetric mass does not need the underlying metric to be smooth nor to approach the Euclidean flat metric. Very heuristically, it measures at large scales the breaking down of the sharp Euclidean isoperimetric inequality; as such it just involves suitable notions of volumes and areas.  I am going to illustrate how the positive mass theorem and the Riemannian Penrose inequality hold in settings that are indeed far from being  smooth and asymptotically flat. The talk is based on results obtained in collaboration with Antonelli, Benatti, Mazzieri, Nardulli and Pozzetta.