Complete Riemannian 4-manifolds with uniformly positive scalar curvature

TL;DR

This paper establishes topological obstructions to complete Riemannian metrics with uniformly positive scalar curvature on certain 4-manifolds, distinguishing standard and exotic smooth structures.

Contribution

It provides the first topological obstructions for complete positive scalar curvature metrics on non-compact 4-manifolds, including exotic ^4's.

Findings

Standard 4-ball distinguished among Mazur manifolds by such metrics

Existence of uncountably many exotic ^4's without such metrics

Any tame 4-manifold has a smooth structure lacking such a metric

Abstract

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.