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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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


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.
  • 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
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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
  • MR Author ID: 335440
  • Email:
  • William P. Minicozzi II
  • Affiliation: Department of Mathematics, Johns Hopkins University, 3400 N. Charles St., Baltimore, Maryland 21218
  • MR Author ID: 358534
  • Email:
  • 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:
  • MathSciNet review: 2020033