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 measures of the boundaries 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. The results about area blow-up sets are used to show that the Allard Regularity Theorems can be applied in some situations where key hypotheses appear to be missing. In particular, we prove a version of the Allard Boundary Regularity Theorem that does not require any area bounds. For example, we prove that if a sequence of smooth minimal submanifolds converge as sets to a subset of a smooth, connected, properly embedded manifold with nonempty boundary, and if the convergence of the boundaries is smooth, then the convergence is smooth everywhere.