A Minimal Lamination of the Unit Ball with Singularities along a Line Segment

TL;DR

This paper constructs a sequence of minimal disks in the unit ball with curvature blow-up along a line segment, revealing detailed singularity structures and extending previous results on curvature blow-up sets.

Contribution

It introduces a new example of minimal disks with curvature blow-up along a line segment, expanding understanding of singularity formation in minimal surface sequences.

Findings

Curvature blows up along a line segment on the vertical axis.

Removable singularities occur along the line segment.

Non-removable singularity at the origin.

Abstract

We construct a sequence of compact embedded minimal disks in the unit ball in Euclidean 3-space whose boundaries are in the boundary of the ball and where the 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 the minimal limit lamination and find removable singularities along the line segment and a non-removable singularity at the origin. This extends a result of Colding and Minicozzi where they constructed a sequence with curvatures blowing up only at the center of the ball, Dean's construction of a sequence with curvatures blowing up at a prescribed discrete set of points, and the classical case of the sequence of re-scaled helicoids with curvatures blowing up along the entire vertical axis.