Bending the helicoid

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.