Potential theory for manifolds with boundary and applications to controlled mean curvature graphs

In this paper we characterize the Neumann-parabolicity of manifolds with boundary in terms of a new form of the classical Ahlfors maximum principle and of a version of the so-called Kelvin-Nevanlinna-Royden criterion. 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 × 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, which extend results by A. Ros and H. Rosenberg valid for domains of ℝ2, and slice-type results for graphs whose superlevel sets have finite volume. Finally, the use of the Ahlfors maximum principle allows us to establish a connection between the Neumann-parabolicity and the Dirichlet-parabolicity commonly used in minimal surface theory. In particular, we will be able to give a deterministic proof of special cases of a result by R. Neel.