Minimal surfaces with closed curvature lines

TL;DR

This paper studies non-orientable minimal surfaces with finite total curvature in three-dimensional space, focusing on those with ends foliated by closed curvature lines, revealing rigidity and non-existence results.

Contribution

It characterizes the rigidity of such minimal surfaces and proves that no surfaces with a single end exist under these conditions.

Findings

No such surfaces with one end exist.

Ends must be foliated by closed lines of curvature.

The condition on ends is necessary for free boundary pieces.

Abstract

We investigate complete non-orientable minimal surfaces of finite total curvature in $\mathbb{R}^3$ such that their ends are foliated by closed lines of curvature. This condition on the ends is necessary if they have a piece inside some Euclidean ball that is free boundary. It turns out this is a rigid situation, and we are able to show, among further obstructions, that there are no such surfaces with one end.