Isoperimetric sets in nonnegative scalar curvature and their role through various concepts of mass

Abstract

We review some recent results about the relations among isoperimetric sets, Penrose inequalities and related concepts in the analysis of $3$-manifolds of nonnegative scalar curvature. We also show that if the isoperimetric sets of big volume have connected boundaries, the equivalence among suitable notions of mass hold.

Final version, to appear in "Springer INdAM series: Anisotropic Isoperimetric Problems & Related Topics"