## Embedded minimal disks: Proper versus nonproper—global versus local

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- by Tobias H. Colding and William P. Minicozzi II PDF
- Trans. Amer. Math. Soc.
**356**(2004), 283-289 Request permission

## 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

- 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.
- —, The space of embedded minimal surfaces of fixed genus in a $3$-manifold IV; Locally simply connected, preprint, math.AP/0210119.
- Robert Osserman,
*A survey of minimal surfaces*, 2nd ed., Dover Publications, Inc., New York, 1986. MR**852409**

## 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
- MR Author ID: 335440
- Email: colding@cims.nyu.edu
**William P. Minicozzi II**- Affiliation: Department of Mathematics, Johns Hopkins University, 3400 N. Charles St., Baltimore, Maryland 21218
- MR Author ID: 358534
- Email: minicozz@jhu.edu
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**356**(2004), 283-289 - MSC (2000): Primary 53A10, 49Q05
- DOI: https://doi.org/10.1090/S0002-9947-03-03230-6
- MathSciNet review: 2020033