Bending the helicoid

Abstract

We construct Colding–Minicozzi limit minimal laminations in open domains in ${\mathbb{R}}^3$ with the singular set of C ¹-convergence being any properly embedded C ^1,1-curve. By Meeks’ C ^1,1-regularity theorem, the singular set of convergence of a Colding–Minicozzi limit minimal lamination ${\mathcal{L}}$ is a locally finite collection $S({\mathcal{L}})$ of C ^1,1-curves that are orthogonal to the leaves of the lamination. Thus, our existence theorem gives a complete answer as to which curves appear as the singular set of a Colding–Minicozzi limit minimal lamination. In the case the curve is the unit circle ${\mathbb{S}}^1(1)$ in the (x ₁, x ₂)-plane, the classical Björling theorem produces an infinite sequence of complete minimal annuli H _n of finite total curvature which contain the circle. The complete minimal surfaces H _n contain embedded compact minimal annuli $\overline{H}_n$ in closed compact neighborhoods N _n of the circle that converge as $n \to \infty$ to $\mathbb {R}^3 - x_3$ -axis. In this case, we prove that the $\overline{H}_n$ converge on compact sets to the foliation of $\mathbb {R}^3 - x_3$ -axis by vertical half planes with boundary the x ₃-axis and with ${\mathbb{S}}^1(1)$ as the singular set of C ¹-convergence. The $\overline{H}_n$ have the appearance of highly spinning helicoids with the circle as their axis and are named bent helicoids.