  Consider a sequence of minimal varieties M_i in a Riemannian manifold N such
that the boundary measures are uniformly bounded on compact sets. Let Z be the
set of points at which the areas of the M_i blow up. We prove that Z behaves in
some ways like a minimal variety without boundary: in particular, it satisfies
the same maximum and barrier principles that a smooth minimal submanifold
satisfies. For suitable open subsets W of N, this allows one to show that if
the areas of the M_i are uniformly bounded on compact subsets of W, then the
areas are in fact uniformly bounded on all compact subsets of N. Similar
results are proved for varieties with bounded mean curvature. Applications
include a version of Allard's boundary regularity theorem in which no mass
bounds are assumed.
