Controlling Area Blow-up in Minimal or Bounded Mean Curvature Varieties

Abstract

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.