Bending the helicoid
Bending the helicoid
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弯曲螺旋面
DOI：
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
Matthias J. Weber
中科院分区：
文献类型：
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作者：
W. H. Meeks;Matthias J. Weber
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 $$n o infty$$ 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.
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 $$n o infty$$ 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.