  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 $\mathbb{R}^4$'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.
