Bending the Helicoid

We construct Colding-Minicozzi limit minimal laminations in open domains in R with the singular set of C-convergence being any properly embedded C-curve. By Meeks’ C-regularity theorem, the singular set of convergence of a Colding-Minicozzi limit minimal lamination L is a locally finite collection S(L) of C-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 S(1) in the (x1, x2)-plane, the classical Björling theorem produces an infinite sequence of complete minimal annuli Hn of finite total curvature which contain the circle. The complete minimal surfaces Hn contain embedded compact minimal annuli Hn in closed compact neighborhoods Nn of the circle that converge as n → ∞ to R − x3-axis. In this case, we prove that the Hn converge on compact sets to the foliation of R 3 − x3-axis by vertical half planes with boundary the x3-axis and with S (1) as the singular set of C-convergence. The Hn have the appearance of highly spinning helicoids with the circle as their axis and are named bent helicoids. Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42