Extending an example by Colding and Minicozzi

TL;DR

This paper constructs a sequence of minimal disks in a cylinder with curvature blow-up at one point, converging to a non-smooth lamination, revealing new behaviors of minimal surfaces.

Contribution

It extends an example by Colding and Minicozzi, demonstrating novel properties of minimal disks and their convergence in Euclidean cylinders.

Findings

Sequence of minimal disks with curvature blow-up

Convergence to a non-smooth minimal lamination

Disks are not properly embedded in exhaustions of R^3

Abstract

Extending an example by Colding and Minicozzi, we construct a sequence of properly embedded minimal disks $\Sigma_i$ in an infinite Euclidean cylinder around the $x_3$-axis with curvature blow-up at a single point. The sequence converges to a non smooth and non proper minimal lamination in the cylinder. Moreover, we show that the disks $\Sigma_i$ are not properly embedded in a sequence of open subsets of $\mathbb{ R}^3$ that exhausts $\mathbb{ R}^3$.