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

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 R^2, 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

Maximum Principles at Infinity and the Ahlfors-Khas’minskii Duality: An Overview

“…4.2] and previous work in [2,28,4,5]. While (37) holds in many instances, there are notable exceptions, for example the eikonal subequation. For such subequations, it is the Ahlfors property the one that actually realizes duality.…”

“…As the cases of parabolicity and stochastic completeness show, the Ahlfors property is also related to the next Liouville one: Indeed, in [54] the main result itself is expressed as a duality between Khas'minskii and Liouville properties. It is not difficult to show that the Ahlfors property implies the Liouville one, and that the two are equivalent provided that (37) u ≡ 0 is F-harmonic, cf. [50,Prop.…”

Height Estimates for Killing Graphs

“…We say that the domain D is smooth if its topological boundary ∂D is a smooth hypersurface Γ with boundary ∂Γ = ∂D ∩ ∂M . Adopting the notation in [16], for any domain D ⊆ M we define…”

“…As usual, the notion of weak supersolution can be obtained by reversing the inequality and, finally, we speak of a weak solution when the equality holds in (16) without any sign condition on ϕ.…”

“…We refer the reader to [16, Theorem 0.9] for a detailed proof of the previous result in the unweighted setting. Although the proof of this theorem can be deduced adapting to the weighted laplacian ∆ f the arguments in [16] (indeed, all results obtained there can be adapted to the weighted case with only minor modifications of the proofs), we provide here a shorter and more elegant argument. In order to do this we will need the following preliminary fact that will be proved without the use of any capacitary argument and, therefore, can be adapted to deal also with the f -parabolicity under Dirichlet boundary conditions; see [19] for old results and recent advances on the Dirichlet parabolicity of (unweighted) Riemannian manifolds.…”