Journal
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY
Volume 48, Issue 3, Pages 325-407Publisher
AMER MATHEMATICAL SOC
DOI: 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
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Funding
- NSF [DMS - 0405836, DMS - 0703213, DMS - 1004003]
- Spanish MEC [MTM2007-61775]
- Regional J. Andalucia [P06-FQM-01642]
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1004003] Funding Source: National Science Foundation
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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 {R-t}(t is an element of(0,1)) of Riemann minimal examples are the only complete, properly embedded, minimal planar domains in R-3; the proof of this result depends primarily on work of Colding and Minicozzi, Collin, Lopez and Ros, Meeks, Perez 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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