Illinois Journal of Mathematics

A minimal lamination of the unit ball with singularities along a line segment

Siddique Khan

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Abstract

We construct a sequence of compact embedded minimal disks in the unit ball in Euclidean 3-space whose boundaries are in the boundary of the ball and where the curvatures blow up at every point of a line segment of the vertical axis, extending from the origin. We further study the transversal structure of the minimal limit lamination and find removable singularities along the line segment and a non-removable singularity at the origin. This extends a result of Colding and Minicozzi where they constructed a sequence with curvatures blowing up only at the center of the ball, Dean’s construction of a sequence with curvatures blowing up at a prescribed discrete set of points, and the classical case of the sequence of re-scaled helicoids with curvatures blowing up along the entire vertical axis.

Article information

Source
Illinois J. Math., Volume 53, Number 3 (2009), 833-855.

Dates
First available in Project Euclid: 4 October 2010

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1286212918

Digital Object Identifier
doi:10.1215/ijm/1286212918

Mathematical Reviews number (MathSciNet)
MR2727357

Zentralblatt MATH identifier
1225.53009

Subjects
Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] 49Q05: Minimal surfaces [See also 53A10, 58E12]

Citation

Khan, Siddique. A minimal lamination of the unit ball with singularities along a line segment. Illinois J. Math. 53 (2009), no. 3, 833--855. doi:10.1215/ijm/1286212918. https://projecteuclid.org/euclid.ijm/1286212918


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References

  • T. H. Colding and W. P. Minicozzi II, Embedded minimal disks: Proper versus nonproper global versus local, Trans. Amer. Math. Soc. 356 (2003), 283–289.
  • T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold I; Estimates off the axis for disks, Ann. of Math. 160 (2004), 27–68.
  • T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold II; Multi-valued graphs in disks, Ann. of Math. 160 (2004), 69–92.
  • T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold III; Planar domains, Ann. of Math. 160 (2004), 523–572.
  • T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold IV; Locally simply connected, Ann. of Math. 160 (2004), 573–615.
  • B. Dean, Embedded minimal disks with prescribed curvature blowup, Proc. Amer. Math. Soc. 134 (2006), 1197–1204.
  • W. H. Meeks III and H. Rosenberg, The uniqueness of the helicoid, Ann. of Math. (2) 161 (2005), 727–758.
  • M. Calle and D. Lee, Non-proper helicoid-like limits of closed minimal surfaces in 3-manifolds, Preprint, available at.
  • D. Hoffman and B. White, Genus-one helicoids from a variational point of view, Comment. Math. Helv. 83 (2008), 767–813.
  • W. H. Meeks III and M. Weber, Bending the helicoid, Math. Ann. 339 (2007), 783–798.
  • R. Osserman, A survey of minimal surfaces, 2nd ed., Dover, New York, 1986.
  • D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer, Berlin, 1983.