The classical theory of minimal surfaces

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 {Rt}t∈(0,1) of Riemann minimal examples are the only complete, properly embedded, minimal planar domains in R{double-struck}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. © 2011 American Mathematical Society Reverts to public domain 28 years from publication. © 2011 American Mathematical Society.