2005/11/15 by William H. Meeks, William H. Meeks III, Meeks, William H. +2
Computer Science · Mathematics · #49Q05 #53A10 #53C42 #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.DG #msc:49Q05 #msc:53A10 #msc:53C42
paper · pdf · doi:10.48550/arxiv.math/0511387
17 pages, 4 figures
arxiv created 2005/11/15 · openalex publication_date 2005/11/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct Colding-Minicozzi limit minimal laminations in open domains in \rth 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 \cal L is a locally finite collection S(\cal 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 \esf1(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 \rth - x3-axis. In this case, we prove that the Hn converge on compact sets to the foliation of \rth - x3-axis by vertical half planes with boundary the x3-axis and with \esf1(1) as the singular set of C1-convergence. The Hn have the appearance of highly spinning helicoids with the circle as their axis and are named \em bent helicoids.