The classical theory of minimal surfaces
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- by William H. Meeks III and Joaquín Pérez PDF
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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.References
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Additional Information
- William H. Meeks III
- Affiliation: Department of Mathematics, University of Massachusetts, Amherst, Massachusetts 01003
- MR Author ID: 122920
- Email: profmeeks@gmail.com
- Joaquín Pérez
- Affiliation: Department of Geometry and Topology, University of Granada, Granada, Spain
- Email: jperez@ugr.es
- 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. - © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2801776