Complete Riemannian 4-manifolds with uniformly positive scalar curvature

We obtain topological obstructions to the existence of a complete Riemannian metric with uniformly positive scalar curvature on certain (non-compact) 4-manifolds. In particular, such a metric on the interior of a compact contractible 4-manifold uniquely distinguishes the standard 4-ball up to diffeomorphism among Mazur manifolds and up to homeomorphism in general. We additionally show there exist uncountably many exotic R⁴'s that do not admit such a metric and that any (non-compact) tame 4-manifold has a smooth structure that does not admit such a metric.