Bending the Helicoid

We construct Colding-Minicozzi limit minimal laminations in open domains in $\text{rth}$ 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 $\mathcal{L}$ 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 ${\text{esf}}^1(1)$ in the $(x_1,x_2)$-plane, the classical Bj"orling 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 $n\to \mathrm{\infty }$ to $\text{rth}-x_3$-axis. In this case, we prove that the ${\bar{H}}_n$ converge on compact sets to the foliation of $\rth - x_3$-axisbyverticalhalfplaneswithboundarythe$x_3$-axisandwith$\esf^1(1)$asthesingularsetof$C^1$-convergence\mathrm{.}The$\bar{H}_n$ have the appearance of highly spinning helicoids with the circle as their axis and are named {\em bent helicoids}.