Comparison geometry for substatic manifolds and a weighted Isoperimetric Inequality

2023/07/27 by Borghini, Stefano, Fogagnolo, Mattia · 4 citations
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2307.14618

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 CD(0, 1) metrics.

Cited by

• Remark on a geometric inequality for closed hypersurfaces in weighted manifolds
• Stability of weighted minimal hypersurfaces under a lower 1-weighted Ricci curvature bound
• A Willmore-type inequality for hypersurfaces with asymptotic or integral Ricci curvature bounds
• Choi-Wang inequality for affine connections

Related

• Rigid comparison geometry for Riemannian bands and open incomplete manifolds
• Isoperimetric problems and lower bounds on curvature
• Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature
• Comparison geometry of manifolds with boundary under a lower weighted Ricci curvature bound
• Rigidities of Isoperimetric inequality under nonnegative Ricci curvature