  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.
