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

TL;DR

This paper studies the topological structure of open simply-connected 3-manifolds with slowly decaying positive scalar curvature, showing that certain exotic manifolds cannot admit such metrics and classifying others topologically.

Contribution

It proves that the Whitehead manifold cannot have a complete metric with slow decay of scalar curvature and classifies certain contractible and non-contractible 3-manifolds under these conditions.

Findings

Whitehead manifold admits no such metric

Contractible complete 3-manifolds with slow decay are homeomorphic to R^3

Manifolds with π₂(M)=Z are homeomorphic to S²×R

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