Extending an example by Colding and Minicozzi

Abstract

Extending an example by Colding and Minicozzi, we construct a sequence of properly embedded minimal disks ${\Sigma }_i$ in an infinite Euclidean cylinder around the $x_3$-axis with curvature blow-up at a single point. The sequence converges to a non smooth and non proper minimal lamination in the cylinder. Moreover, we show that the disks ${\Sigma }_i$ are not properly embedded in a sequence of open subsets of ${\mathbb{R}}^3$ that exhausts ${\mathbb{R}}^3$.

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