  In this paper we extend to non-compact Riemannian manifolds with boundary the
use of two important tools in the geometric analysis of compact spaces, namely,
the weak maximum principle for subharmonic functions and the integration by
parts. The first one is a new form of the classical Ahlfors maximum principle
whereas the second one is a version for manifolds with boundary of the so
called Kelvin-Nevanlinna-Royden criterion of parabolicity. In fact, we will
show that the validity of non-compact versions of these tools serve as a
characterization of the Neumann parabolicity of the space.
  The motivation underlying this study is to obtain new information on the
geometry of graphs with prescribed mean curvature inside a Riemannian product
of the type $N\times\mathbb{R}$. In this direction two kind of results will be
presented: height estimates for constant mean curvature graphs parametrized
over unbounded domains in a complete manifold and slice type results for graphs
whose superlevel sets have finite volume.
