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Any sequence of properly embedded minimal disks in an open subset U ofEuclidean 3-space has a subsequence such that the curvatures blow up on arelatively closed subset K of U and such that the disks converge in thecomplement of K to a minimal lamination of U\K. Assuming results ofColding-Minicozzi and an extension due to Meeks, we prove that such a blow-upset K must be contained in a C^1 embedded curve.

Curvatures of embedded minimal disks blow up on subsets of C^1 curves