A Minimal Lamination with Cantor Set-Like Singularities

TL;DR

This paper constructs a sequence of minimal surfaces converging to a lamination with singularities concentrated on a Cantor set-like boundary, revealing detailed curvature blow-up and non-removable singularities.

Contribution

It introduces a novel method to create minimal laminations with prescribed singular sets resembling Cantor sets, advancing understanding of singularities in minimal surface theory.

Findings

Curvature blows up exactly on the singular set M

The limit lamination has non-removable singularities on M's boundary

Constructs explicit examples of minimal laminations with Cantor set-like singularities

Abstract

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