Solution of the problem of Plateau
Jesse Douglas · Transactions of the American Mathematical Society · 1931
The problem of Plateau is to prove the existence of a minimal surface bounded by a given contour.This memoir presents the first solution of this problem for the most general kind of contour: an arbitrary Jordan curve in «-dimensional euclidean space.Topological complications in the contour, as well as the dimensionality n of the containing space, are without consequence for either method or result.Naturally, an arrangement of knots in the contour will produce corresponding complications in the minimal surface, such as self-intersections and branch points.The method used is entirely novel, representing a complete departure from the classical modes of attack hitherto employed.In this introduction we shall outline three of the classic methods (wherein n is always 3) and, fourth, the method of the present paper, which we believe to furnish the key to the problem.That this is the fact will become even clearer when, in future papers, we apply this method to the case of several contours and of various topological structures of the minimal surface, f for instance, a Möbius leaf with a prescribed boundary.It is to be signalized that the solution here given is strictly elementary, employing only the most simple and usual parts of analysis, and that the presentation is self-sufficient, requiring no special preliminary knowledge.(1) First to be considered, in this introductory survey, is the method based on the ideas of Riemann, Weierstrass and Schwarz.fHere the given * This work, in successive stages of its development, was presented to the Society at various meetings from December, 1926, to December, 1929.