最小或有界中曲率簇的控制区爆破
Controlling Area Blow-up in Minimal or Bounded Mean Curvature Varieties
摘要：
考虑黎曼流形N中的极小簇序列M_i，使得边界测度在紧集上一致有界.设Z是M_i的面积在其上爆炸的点的集合。我们证明了Z在某些方面表现得像一个没有边界的极小簇：特别地，它满足与光滑极小子流形相同的极大值原理和障碍原理。对于N的适当开子集W，这允许我们证明：如果M_i的面积在W的紧子集上一致有界，那么这些面积实际上在N的所有紧子集上一致有界。对于有界中曲率的簇也证明了类似的结果。应用程序包括一个版本的阿拉德的边界正则性定理，其中没有质量界限假设。
英文摘要：
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.
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