3-Manifolds with positive scalar curvature and bounded geometry

We show that a complete contractible 3-manifold with positive scalar curvature and bounded geometry must be ${\mathbb{R}}^3$. We also show that an open handlebody of genus larger than 1 does not admit complete metrics with positive scalar curvature and bounded geometry. Our results rely on the maximal weak solution to inverse mean curvature flow due to the third-named author.