Comparison geometry for substatic manifolds and a weighted Isoperimetric Inequality

Abstract

Substatic Riemannian manifolds with minimal boundary arise naturally in General Relativity as spatial slices of static spacetimes satisfying the Null Energy Condition. Moreover, they constitute a vast generalization of nonnegative Ricci curvature. In this paper we will prove various geometric results in this class, culminating in a sharp, weighted Isoperimetric inequality that quantifies the area minimizing property of the boundary. Its formulation and proof will build on a comparison theory partially stemming from a newly discovered conformal connection with $\mathrm{CD}(0, 1)$ metrics.