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

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.

A Minimal Lamination of the Interior of a Positive Cone with Quadratic Curvature Blowup

“…We seek a graph over G by a function w(s, θ) of the form w(s, θ) = e δθ u(s), (7) so that H w ≡ 0. Here we have denoted by H w the mean curvature of G w , the normal graph over G by w:…”

“…It remains to show whether or not such laminations exist for rates between quadratic and quartic. For other, more pathological examples of laminations of open regions of B 1 by embedded minimal disks see [4,5,7,8].…”

Logarithmically spiraling helicoids

“…The second author, in [12], uses the Weierstrass representation to prove the same result as in [6]. For additional constructions in the same spirit, see also [3,4,11].…”

A minimal lamination with Cantor set-like singularities

“…Later, B. Dean in [3] found a similar example where the limit lamination has a finite set of singularities along a line segment, and S. Khan in [4] found a limit lamination consisting of a non-properly embedded minimal disk in the upper half ball spiraling into a foliation by parallel planes of the lower half ball. Both Dean and Khan used methods that are analogous to those in [1].…”