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 ${\mathbb{S}}^2\times \mathbb{R}$.