Distortions of the Helicoid

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 $R$.