Mathematische Annalen

, Volume 339, Issue 4, pp 783–798

Bending the helicoid

Article

DOI: 10.1007/s00208-007-0120-4

Cite this article as:
Meeks, W.H. & Weber, M. Math. Ann. (2007) 339: 783. doi:10.1007/s00208-007-0120-4

Abstract

We construct Colding–Minicozzi limit minimal laminations in open domains in \({\mathbb{R}}^3\) with the singular set of C1-convergence being any properly embedded C1,1-curve. By Meeks’ C1,1-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 C1,1-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 (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 \(\overline{H}_n\) in closed compact neighborhoods Nn 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 x3-axis and with \({\mathbb{S}}^1(1)\) as the singular set of C1-convergence. The \(\overline{H}_n\) have the appearance of highly spinning helicoids with the circle as their axis and are named bent helicoids.

Mathematics Subject Classification(2000)

Primary 53A10 Secondary 49Q05 53C42 

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Math DepartmentUniversity of MassachusettsAmherstUSA
  2. 2.Math DepartmentUniversity of IndianaBloomingtionUSA

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