Capacity for minimal graphs over manifolds and the half-space property

TL;DR

This paper introduces a capacity concept for minimal graphs over manifolds, explores the notion of M-parabolicity, and establishes boundary behavior, existence results, and a half-space theorem for minimal hypersurfaces in product manifolds.

Contribution

It defines M-capacities and M-parabolicity, linking them to minimal hypersurfaces and boundary behavior, and proves a half-space theorem extending classical results.

Findings

M-parabolicity implies boundary regularity for minimal graphs.

Half-space theorem holds for M-parabolic manifolds in product spaces.

M-parabolic ends are parabolic if Ricci curvature is bounded below.

Abstract

In this paper, we define natural capacities using a relative volume of graphs over manifolds, which can be characterized by solutions of bounded variation to Dirichlet problems of minimal hypersurface equation. Using the capacities, we introduce a notion '$M$-parabolicity' for ends of complete manifolds, where a parabolic end must be $M$-parabolic, but not vice versa in general. We study the boundary behavior of solutions associated with capacities in the measure sense, and the existence of minimal graphs over $M$-parabolic or $M$-nonparabolic manifolds outside compact sets. For a $M$-parabolic manifold $P$, we prove a half-space theorem for complete proper minimal hypersurfaces in $P\times\mathbb{R}$. As a corollary, we immediately have a slice theorem for smooth mean concave domains in $P\times\mathbb{R}^+$, where the $M$-parabolic condition is sharp by our example. On the other hand, we prove that any $M$-parabolic end is indeed parabolic provided its Ricci curvature is uniformly bounded from below. Compared to harmonic functions, we get the asymptotic estimates with sharp orders for minimal graphic functions on nonparabolic manifolds of nonnegative Ricci curvature outside compact sets.