Distortions of the Helicoid

TL;DR

This paper investigates the local geometric structure of embedded minimal disks in three-dimensional space, demonstrating that while they resemble helicoids at small scales, this similarity does not extend to larger scales.

Contribution

It refines the understanding of minimal disks by showing the limitations of helicoid approximation at larger scales, contrasting previous results on local behavior.

Findings

Minimal disks resemble helicoids at small scales.

Helicoid approximation fails at larger scales.

Provides insights into the scale-dependent geometry of minimal surfaces.

Abstract

Colding and Minicozzi have shown that an embedded minimal disk $0\in\Sigma\subset B_R$ in $\Real^3$ with large curvature at 0 looks like a helicoid on the scale of $R$. Near 0, this can be sharpened: on the scale of $|A|^{-1}(0)$, $\Sigma$ is close, in a Lipschitz sense, to a piece of a helicoid. We use surfaces constructed by Colding and Minicozzi to see this description cannot hold on the scale $R$.