Uniformly Positive Scalar Curvature on Manifold Interiors

We obtain topological obstructions to the existence of a complete Riemannian metric with uniformly positive scalar curvature on the interiors of certain 4-manifolds using Seiberg–Witten theory. In particular, no punctured blowups of a homotopy K3 manifold admit complete metrics of uniformly positive scalar curvature. Further, let $X$ be a compact oriented smooth manifold of dimension $5$, $6$, or $7$. We show that if $\mathrm{Int}X$ admits a complete metric of uniformly positive scalar curvature, then $X$ admits a positive scalar curvature metric with minimal boundary. In dimensions $6$ and $7$, the metric may moreover be chosen to be a product near the boundary. In dimension 5, under additional hypotheses and after possibly changing the smooth structure on $X$, one can likewise obtain a positive scalar curvature metric that is a product near the boundary.