Comparison geometry for substatic manifolds and a weighted Isoperimetric Inequality

TL;DR

This paper develops a comparison geometry framework for substatic manifolds, leading to a sharp weighted isoperimetric inequality that extends classical results and has implications in General Relativity.

Contribution

It introduces a new comparison theory for substatic manifolds using a conformal connection related to D(0,1) metrics, resulting in a novel weighted isoperimetric inequality.

Findings

Established a sharp weighted isoperimetric inequality for substatic manifolds.

Connected substatic geometry with D(0,1) metrics via a conformal approach.

Extended classical geometric inequalities to a broader class of manifolds.

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.