Contractible 3-manifolds and positive scalar curvature

It is not known whether a contractible 3-manifold admits a complete metric of positive scalar curvature. For example, the Whitehead manifold is a contractible 3-manifold but not homeomorphic to $R^{3}$. In this talk, I will present my proof that it does not have a complete metric with positive scalar curvature. I will further explain that a contractible genus one 3-manifold, a notion introduced by McMillan, does not admit a complete metric of positive scalar curvature.