A Minimal Lamination with Cantor Set-Like Singularities

Given a compact closed subset $M$ of a line segment in ${\mathbb{R}}^3$, we construct a sequence of minimal surfaces ${\mathrm{\Sigma }}_k$ 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 $M$.