Structure theorems for singular minimal laminations

TL;DR

This paper establishes global structure theorems for possibly singular minimal laminations in three-dimensional space, providing foundational results for understanding the geometry and topology of minimal surfaces.

Contribution

It introduces two new global structure theorems for singular minimal laminations using local removable singularity and topology scale theorems.

Findings

Descriptive results for singular minimal laminations in ^3

Applications to bounds on index and ends of minimal surfaces

Proof of properness for certain minimal surfaces with finite genus

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.