Bending the Helicoid

Abstract

We construct Colding-Minicozzi limit minimal laminations in open domains in $\rth$ 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 ${\cal L}$ is a locally finite collection $S({\cal L})$ 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 $\esf^1(1)$ 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 $\bar{H}_n$ in closed compact neighborhoods $N_n$ of the circle that converge as $n \to \infty$ to $\rth - x_3$-axis. In this case, we prove that the $\bar{H}_n$ converge on compact sets to the foliation of $\rth - x_3$-axis by vertical half planes with boundary the $x_3$-axis and with $\esf^1(1)$ as the singular set of $C^1$-convergence. The $\bar{H}_n$ have the appearance of highly spinning helicoids with the circle as their axis and are named {\em bent helicoids}.