Topology of 3-manifolds with nonnegative scalar curvature and positive harmonic functions

We study complete $3$-manifolds with nonnegative scalar curvature under additional regularity assumptions. We prove that a contractible such manifold is diffeomorphic to ${\mathbb{R}}^3$, and that an open handlebody admitting such a metric must have genus at most $1$. The proof uses exhaustions by level sets of harmonic functions and refined average gradient estimates.