Simply-connected open 3-manifolds with slow decay of positive scalar curvature

Abstract

The goal of this paper is to investigate the topological structure of open simply-connected 3-manifolds whose scalar curvature has a slow decay at infinity. In particular, we show that the Whitehead manifold does not admit a complete metric, whose scalar curvature decays slowly, and in fact that any contractible complete 3-manifolds with such a metric is homeomorphic to $\mathbb{R}^{3}$. Furthermore, using this result, we prove that any open simply-connected 3-manifold $M$ with $\pi_{2}(M)=\mathbb{Z}$ and a complete metric as above, is homeomorphic to $\mathbb{S}^{2}\times \mathbb{R}$.