Embedded minimal disks with prescribed curvature blowup
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- by Brian Dean PDF
- Proc. Amer. Math. Soc. 134 (2006), 1197-1204 Request permission
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.References
- Tobias H. Colding and William P. Minicozzi II, Embedded minimal disks: proper versus nonproper—global versus local, Trans. Amer. Math. Soc. 356 (2004), no. 1, 283–289. MR 2020033, DOI 10.1090/S0002-9947-03-03230-6
- 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, preprint, math.AP/0210119.
- W. Meeks and M. Weber, in preparation.
- Robert Osserman, A survey of minimal surfaces, 2nd ed., Dover Publications, Inc., New York, 1986. MR 852409
Additional Information
- Brian Dean
- Affiliation: Department of Mathematics, Hylan Building, University of Rochester, Rochester, New York 14627
- Email: bdean@math.rochester.edu
- 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
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 1197-1204
- MSC (2000): Primary 53C42; Secondary 53A10, 57R40
- DOI: https://doi.org/10.1090/S0002-9939-05-08045-7
- MathSciNet review: 2196057