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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Embedded minimal disks with prescribed curvature blowup


Author: Brian Dean
Journal: Proc. Amer. Math. Soc. 134 (2006), 1197-1204
MSC (2000): Primary 53C42; Secondary 53A10, 57R40
Published electronically: July 20, 2005
MathSciNet review: 2196057
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Abstract: We construct a sequence of compact embedded minimal disks in a ball in $\mathbb{R} ^3$, whose boundaries lie in the boundary of the ball, such that the curvature blows up only at a prescribed discrete (and hence, finite) set of points on the $x_3$-axis. This extends a result of Colding and Minicozzi, who constructed a sequence for which the curvature blows up only at the center of the ball, and is a partial affirmative answer to the larger question of the existence of a sequence for which the curvature blows up precisely on a prescribed closed set on the $x_3$-axis.


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Additional Information

Brian Dean
Affiliation: Department of Mathematics, Hylan Building, University of Rochester, Rochester, New York 14627
Email: bdean@math.rochester.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08045-7
PII: S 0002-9939(05)08045-7
Received by editor(s): August 10, 2004
Received by editor(s) in revised form: October 26, 2004
Published electronically: July 20, 2005
Additional Notes: The author thanks W. Minicozzi for his many helpful discussions.
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.