Positive scalar curvature with skeleton singularities

TL;DR

This paper investigates how certain singularities in Riemannian manifolds affect the existence of positive scalar curvature, showing that specific edge and point singularities do not alter the Yamabe type and establishing positive mass theorems in these contexts.

Contribution

It extends scalar curvature comparison theory to manifolds with skeleton singularities, demonstrating invariance of Yamabe type under certain edge and point singularities and deriving positive mass theorems.

Findings

Edge singularities with cone angles ≤ 2π do not affect Yamabe type.

In three dimensions, more general singular sets also do not affect Yamabe type.

Positive Mass Theorems are established for manifolds with these singularities.

Abstract

We study positive scalar curvature on the regular part of Riemannian manifolds with singular, uniformly Euclidean ($L^\infty$) metrics that consolidate Gromov's scalar curvature polyhedral comparison theory and edge metrics that appear in the study of Einstein manifolds. We show that, in all dimensions, edge singularities with cone angles $\leq 2\pi$ along codimension-2 submanifolds do not affect the Yamabe type. In three dimensions, we prove the same for more general singular sets, which are allowed to stratify along 1-skeletons, exhibiting edge singularities (angles $\leq 2\pi$) and arbitrary $L^\infty$ isolated point singularities. We derive, as an application of our techniques, Positive Mass Theorems for asymptotically flat manifolds with analogous singularities.