Structure theorems for singular minimal laminations

Abstract

We apply the local removable singularity theorem for minimal laminations and the local picture theorem on the scale of topology to obtain two descriptive results for certain possibly singular minimal laminations of $\mathbb{R}^3$. These two global structure theorems will be applied in forthcoming papers to obtain bounds on the index and the number of ends of complete, embedded minimal surfaces of fixed genus and finite topology in $\mathbb{R}^3$, and to prove that a complete, embedded minimal surface in $\mathbb{R}^3$ with finite genus and a countable number of ends is proper.