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We construct a sequence of compact embedded minimal disks in the unit ball inEuclidean 3-space whose boundaries are in the boundary of the ball and wherethe curvatures blow up at every point of a line segment of the vertical axis,extending from the origin. We further study the transversal structure of theminimal limit lamination and find removable singularities along the linesegment and a non-removable singularity at the origin. This extends a result ofColding and Minicozzi where they constructed a sequence with curvatures blowingup only at the center of the ball, Dean's construction of a sequence withcurvatures blowing up at a prescribed discrete set of points, and the classicalcase of the sequence of re-scaled helicoids with curvatures blowing up alongthe entire vertical axis.

A minimal lamination of the unit ball with singularities along a line segment