Global maximum principles and divergence theorems on complete manifolds with boundary

TL;DR

This paper extends maximum principles and divergence theorems to non-compact manifolds with boundary, providing new tools for geometric analysis and insights into the geometry of mean curvature graphs in Riemannian products.

Contribution

It introduces non-compact versions of maximum principles and divergence theorems, characterizing Neumann parabolicity and applying these to geometric problems involving mean curvature graphs.

Findings

Characterization of Neumann parabolicity via maximum principles

Height estimates for constant mean curvature graphs

Slice type results for graphs with finite volume superlevel sets

Abstract

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.