Positive mass and isoperimetry for continuous metrics with nonnegative scalar curvature

Abstract

This paper deals with quasi-local isoperimetric versions of the positive mass theorem on $3$-manifolds endowed with continuous complete metrics having nonnegative scalar curvature in a suitable weak sense. As a corollary, we derive existence results for isoperimetric sets in this low regularity setting. Our main tool is a new local version of the weak inverse mean curvature flow enjoying $C^0$-stable quantitative estimates.