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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Embedded minimal disks: Proper versus nonproper--global versus local


Authors: Tobias H. Colding and William P. Minicozzi II
Journal: Trans. Amer. Math. Soc. 356 (2004), 283-289
MSC (2000): Primary 53A10, 49Q05
Published electronically: August 25, 2003
MathSciNet review: 2020033
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Abstract | References | Similar Articles | Additional Information

Abstract: We construct a sequence of compact embedded minimal disks in a ball in $\mathbf{R}^3$ with boundaries in the boundary of the ball and where the curvatures blow up only at the center. The sequence converges to a limit which is not smooth and not proper. If instead the sequence of embedded disks had boundaries in a sequence of balls with radii tending to infinity, then we have shown previously that any limit must be smooth and proper.


References [Enhancements On Off] (What's this?)

  • [CM1] T.H. Colding and W.P. Minicozzi II, Embedded minimal disks, To appear in The Proceedings of the Clay Mathematics Institute Summer School on the Global Theory of Minimal Surfaces. MSRI. math.DG/0206146.
  • [CM2] -, The space of embedded minimal surfaces of fixed genus in a $3$-manifold IV; Locally simply connected, preprint, math.AP/0210119.
  • [Os] Robert Osserman, A survey of minimal surfaces, 2nd ed., Dover Publications, Inc., New York, 1986. MR 852409 (87j:53012)

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Additional Information

Tobias H. Colding
Affiliation: Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012 and Princeton University, Fine Hall, Washington Rd., Princeton, New Jersey 08544-1000
Email: colding@cims.nyu.edu

William P. Minicozzi II
Affiliation: Department of Mathematics, Johns Hopkins University, 3400 N. Charles St., Baltimore, Maryland 21218
Email: minicozz@jhu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03230-6
PII: S 0002-9947(03)03230-6
Received by editor(s): October 21, 2002
Published electronically: August 25, 2003
Additional Notes: The authors were partially supported by NSF Grants DMS 0104453 and DMS 0104187
Article copyright: © Copyright 2003 American Mathematical Society