A minimal lamination with Cantor set-like singularities

Abstract

Abstract. Given a compact closed subset M of a line segment in R 3 , we construct a sequence of minimal surfaces Σ k embedded in a neighborhood C of the line segment that converge smoothly to a limit lamination of C away from M . Moreover, the curvature of this sequence blows up precisely on M , and the limit lamination has non-removable singularities precisely on the boundary of M .

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

“…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].…”

Helicoid-like minimal disks and uniqueness

“…Theorem 1.1 sharpens this description, though on a much smaller scale, giving a quantitative description of such a disk near a point of large curvature. Examples constructed by Colding and Minicozzi [5], Khan [20], Kleene [21], and Hoffman and White [17] demonstrate that this sharper description cannot hold on the outer scale R -we refer the interested reader to [2].…”

Extending an Example by Colding and Minicozzi

“…Following this example, a number of similar results have been obtained via analogous methods, in which comparable wild limits are observed with the curvature blowing up at a finite set of points on a line [Dea06], along a closed segment [Kha09], or more generally any compact subset of a line [Kle12]. Using variational methods, Hoffman and White [HW11] provided examples of minimal disks in an infinite Euclidean cylinder with curvature blow-ups along any prescribed compact subset of the axis of the cylinder.…”