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

TL;DR

This paper investigates the behavior of sequences of minimal and bounded mean curvature varieties, showing that their blow-up sets behave like minimal varieties and establishing boundary regularity results without mass bounds.

Contribution

It demonstrates that blow-up sets of such varieties satisfy maximum and barrier principles, extending boundary regularity theorems without requiring mass bounds.

Findings

Blow-up sets behave like minimal varieties without boundary.

Uniform area bounds on subsets imply bounds on entire manifolds.

Extended Allard's boundary regularity theorem without mass bounds.

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.