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

TL;DR

This paper proves that the set where curvature blows up for sequences of embedded minimal disks in 3-space must lie within a C^1 curve, refining understanding of their geometric singularities.

Contribution

It establishes that the blow-up set of curvatures in minimal disks is contained in a C^1 embedded curve, extending previous results with new geometric constraints.

Findings

Blow-up sets are contained in C^1 curves.

Sequences of minimal disks converge outside the blow-up set to a lamination.

The result relies on extending known theorems by Colding-Minicozzi and Meeks.

Abstract

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