In this article, we are interested in the question whether any complete contractible $3$-manifold of positive scalar curvature is homeomorphic to $\mathbb{ R}^{ 3}$. We study the fundamental group at infinity, $\pi_{ 1}^{ \infty}$, and its relationship with the existence of complete metrics of positive scalar curvature. We prove that a complete contractible $3$-manifold with positive scalar curvature and trivial $\pi^{ \infty}_{ 1}$ is homeomorphic to $\mathbb{ R}^{ 3}$.
Categories: math.DG math.GT