Some Singular Limit Laminations of Embedded Minimal Planar Domains

In this paper we give two examples of sequences of embedded minimal planar domains in $\mathbb{R}^3$ which converge to singular laminations of $\mathbb{R}^3$. In contrast with the situation for embedded minimal disks, these examples do not arise from complete embedded minimal planar domains and highlight some of the subtleties inherent in understanding refined properties of embedded minimal planar domains.