douglas - MATHEMATICAL PERSPECTIVES BULLETIN(New Series OF...

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Unformatted text preview: MATHEMATICAL PERSPECTIVES BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 45, Number 2, April 2008, Pages 293–302 S 0273-0979(08)01192-0 Article electronically published on February 14, 2008 ABOUT THE COVER: THE WORK OF JESSE DOUGLAS ON MINIMAL SURFACES JEREMY GRAY AND MARIO MICALLEF This paper is dedicated to the memory of our student, Adam Merton (1920- 1999), without whose interest in minimal surfaces and the work of Jesse Douglas we would never have embarked on this study. 1. Introduction The demonstration of the existence of a least-area surface spanning a given contour has a long history. Through the soap-film experiments of Plateau, this existence problem became known as the Plateau problem. By the end of the 19th century, the list of contours for which the Plateau problem could be solved was intriguingly close to that considered by Plateau, namely polygonal contours, lines or rays extending to infinity and circles. Free boundaries on planes were also con- sidered, but even these contours were required to have some special symmetry; for example a pair of circles had to be in parallel planes. These solutions were obtained by finding the pair of holomorphic functions (which, for the experts, we shall refer to as the conformal factor f and the Gauss map g ) required in the so- called Weierstrass-Enneper representation of a minimal surface. In 1928, Garnier completed a programme started independently by Riemann and Weierstrass in the 1860s and then continued by Darboux towards the end of the 19th century for find- ing f and g for an arbitrary polygonal contour. In his 92-page paper [8] (which is very difficult to read and which apparently has never been fully checked) Garnier also employed a limiting process to extend his solution of the Plateau problem to contours which consisted of a finite number of unknotted arcs with bounded cur- vature. Garnier’s work was soon eclipsed by the works of Rad´ o and Jesse Douglas, Received by the editors June 11, 2007, and, in revised form, August 20, 2007. 2000 Mathematics Subject Classification. Primary 01A60, 53A10, 58E12. c ° 2008 American Mathematical Society Reverts to public domain 28 years from publication 293 294 JEREMY GRAY AND MARIO MICALLEF about which there is a considerable amount of inaccurate information in the lit- erature and on which we hope to shed some light in this article. In particular, we challenge the popular belief that Douglas arrived at his functional for solving the Plateau problem by direct consideration of Dirichlet’s integral and its relation to the area functional. Douglas was awarded one of the first Fields Medals for his work on the Plateau problem. There are many amusing aspects of the Fields Medal ceremony at which Douglas was awarded his prize. We simply mention that Wiener collected the medal on behalf of Douglas, even though Douglas did attend the International Congress. And in the address, Carath´ eodory described a method for finding a minimal surface that is due to Rad´...
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douglas - MATHEMATICAL PERSPECTIVES BULLETIN(New Series OF...

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