Calabi-Yau type theorem for complete manifolds with nonnegative scalar curvature

In this paper, we are able to prove an analogy of the Calabi-Yau theorem for complete Riemannian manifolds with nonnegative scalar curvature which are aspherical at infinity. The key tool is an existence result for arbitrarily large bounded regions with weakly mean-concave boundary in Riemannian manifolds with sublinear volume growth. As an application, we use the same tool to show that a complete contractible Riemannian $3$-manifold with positive scalar curvature and sublinear volume growth is necessarily homeomorphic to ${\mathbb{R}}^3$.