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

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.