The scalar curvature in conical manifolds: some results on existence and obstructions

TL;DR

This paper extends scalar curvature existence results to manifolds with isolated conical singularities and explores obstructions using index theory and K-area, advancing understanding of scalar curvature in singular spaces.

Contribution

It generalizes classical scalar curvature existence theorems to conical manifolds and introduces obstructions via index theory and K-area in this setting.

Findings

Existence of scalar curvature metrics on conical manifolds under certain conditions.

Extension of Kazdan-Warner and Cruz-Vitorio results to singular spaces.

Identification of obstructions using index theory and K-area.

Abstract

We first show that existence results due to Kazdan-Warner and Cruz-Vit\'orio can be extended to the category of manifolds with an isolated conical singularity. More precisely, we check that, under suitable conditions on the link manifold, any bounded and smooth function which is negative somewhere is the scalar curvature of some conical metric (with the boundary being minimal whenever it is non-empty). By way of comparison, we complement this analysis by indicating how index theory, as developed by Albin-Gell-Redman, may be used to transfer to this conical setting some of the classical obstructions to the existence of metrics with positive scalar curvature in the spin context. In particular, we use a version of the notion of infinite $K$-area to obstruct such metrics.