Some Singular Limit Laminations of Embedded Minimal Planar Domains

TL;DR

This paper presents two examples of embedded minimal planar domains in three-dimensional space that converge to singular laminations, revealing complexities not seen in complete embedded minimal planar domains.

Contribution

The paper introduces novel examples of embedded minimal planar domains converging to singular laminations, contrasting with known cases from complete domains.

Findings

Examples of convergence to singular laminations

Differences from complete embedded minimal domains

Insights into properties of embedded minimal planar domains

Abstract

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.