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

DOI:
https://doi.org/10.1090/S0002-9947-03-03230-6

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 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.

**[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 -manifold IV; Locally simply connected, preprint, math.AP/0210119.**[Os]**R. Osserman, A survey of minimal surfaces,*Dover*, 2nd. edition (1986). MR**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:
https://doi.org/10.1090/S0002-9947-03-03230-6

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