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.