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.

Potential theory for manifolds with boundary and applications to controlled mean curvature graphs / Impera, Debora; Pigola, Stefano; Setti, Alberto G.. - In: JOURNAL FÜR DIE REINE UND ANGEWANDTE MATHEMATIK. - ISSN 0075-4102. - STAMPA. - 733:0(2017), pp. 121-159. [10.1515/crelle-2014-0137]

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

Impera, Debora;
2017

Abstract

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2691129