Bending the helicoid

Abstract

We construct Colding-Minicozzi limit minimal laminations in open domains in 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 L is a locally finite collection S(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 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 H n in closed compact neighborhoods N n of the circle that converge as n → ∞ to R 3 − x 3 -axis. In this case, we prove that the H n converge on compact sets to the foliation of R 3 − x 3 -axis by vertical half planes with boundary the x 3 -axis and with S 1 (1) as the singular set of C 1 -convergence. The H n have the appearance of highly spinning helicoids with the circle as their axis and are named bent helicoids.

The classical theory of minimal surfaces

“…Discovered by Meeks and Weber [140] and independently by Mira [149], these are complete, immersed minimal annuli H n ⊂ R 3 with two non-embedded ends and finite total curvature; each of the surfaces H n contains the unit circle S 1 in the (x 1 , x 2 )-plane, and a neighborhood of S 1 in H n contains an embedded annulus H n which approximates, for n large, a highly spinning helicoid whose usual straight axis has been periodically bent into the unit circle S 1 (thus the name of bent helicoids); see Figure 2, right. Furthermore, the H n converge as n → ∞ to the foliation of R 3 minus the x 3 -axis by vertical half-planes with boundary the x 3 -axis, and with S 1 as the 1 that spins an arbitrary number n of times around the circle.…”

“…This construction also makes sense when n is half an integer; in the case n = 1 2 , H 1/2 is the double cover of the Meeks minimal Möbius strip described in the previous example. The bent helicoids H n play an important role in proving the converse of Meeks' C 1,1 -Regularity Theorem (Theorem 9.6 below) for the singular set of convergence in a Colding-Minicozzi limit minimal lamination (for this converse, see Meeks and Weber [140]). …”

On Chasles' Property of the Helicoid in Tri-Twisted Real Ambient Space

“…Bent helicoids along a generating curve have been studied quite a lot recently. Although we do not aim here to summarize any recent developments, in support to our claim we would like to remind just one direction of study, since bent helicoids along a circle play a central role in [17] as well as in [1]. This idea will motivate our Example 2.…”

New Examples of Maximal Surfaces in Lorentz-Minkowski Space

“…(1) Comparing with the bending helicoids in Euclidean space [11], and for small values of v, our examples give embedded strips of maximal surfaces only α is not a full circle, that is, only when α is a piece of length less than 2π because V (t) is not a periodic function. A way to get a periodic vector field V (t) is replacing the function ϕ(t) in (2) by a periodic function, as for example, ϕ(t) = cos(t).…”