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The classical theory of minimal surfaces


Authors: William H. Meeks III and Joaquín Pérez
Journal: Bull. Amer. Math. Soc. 48 (2011), 325-407
MSC (2010): Primary 53A10; Secondary 49Q05, 53C42
DOI: https://doi.org/10.1090/S0273-0979-2011-01334-9
Published electronically: March 25, 2011
MathSciNet review: 2801776
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Abstract: We present here a survey of recent spectacular successes in classical minimal surface theory. We highlight this article with the theorem that the plane, the helicoid, the catenoid and the one-parameter family $ \{\mathcal{R}_t\}_{t\in (0,1)}$ of Riemann minimal examples are the only complete, properly embedded, minimal planar domains in $ \mathbb{R}^3$; the proof of this result depends primarily on work of Colding and Minicozzi, Collin, López and Ros, Meeks, Pérez and Ros, and Meeks and Rosenberg. Rather than culminating and ending the theory with this classification result, significant advances continue to be made as we enter a new golden age for classical minimal surface theory. Through our telling of the story of the classification of minimal planar domains, we hope to pass on to the general mathematical public a glimpse of the intrinsic beauty of classical minimal surface theory and our own perspective of what is happening at this historical moment in a very classical subject.


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Additional Information

William H. Meeks III
Affiliation: Department of Mathematics, University of Massachusetts, Amherst, Massachusetts 01003
Email: profmeeks@gmail.com

Joaquín Pérez
Affiliation: Department of Geometry and Topology, University of Granada, Granada, Spain
Email: jperez@ugr.es

DOI: https://doi.org/10.1090/S0273-0979-2011-01334-9
Keywords: Minimal surface, minimal lamination, locally simply connected, finite total curvature, conformal structure, harmonic function, recurrence, transience, parabolic Riemann surface, harmonic measure, universal superharmonic function, Jacobi function, stability, index of stability, Shiffman function, Korteweg-de Vries equation, KdV hierarchy, algebro-geometric potential, curvature estimates, maximum principle at infinity, limit tangent plane at infinity, parking garage, minimal planar domain.
Received by editor(s): September 4, 2006
Received by editor(s) in revised form: October 15, 2007, November 30, 2009, and February 2, 2011
Published electronically: March 25, 2011
Additional Notes: This material is based upon work for the NSF under Awards No. DMS - 0405836, DMS - 0703213, DMS - 1004003. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the NSF
Research partially supported by a Spanish MEC-FEDER Grant no. MTM2007-61775 and a Regional J. Andalucía Grant no. P06-FQM-01642.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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