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

摘要

In this paper we characterize the Neumann-parabolicity of manifolds with boundaryin terms of a new form of the classical Ahlfors maximum principle and ofa version of the so-called Kelvin–Nevanlinna–Royden criterion.The motivation underlying this study is to obtain new information on thegeometry of graphs with prescribed mean curvature inside a Riemannian productof the type N × ? {Nimesmathbb{R}} . In this direction two kind of results will bepresented: height estimates for constant mean curvature graphs parametrizedover unbounded domains in a complete manifold, which extend results by A. Ros and H. Rosenbergvalid for domains of ? 2 {mathbb{R}^{2}} , and slice-type results for graphs whose superlevelsets 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.