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On the problem of Plateau. (English) Zbl 0007.11804
Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 2, No. 2. Berlin: Julius Springer. 109 S., 1 Fig. (1933).
Since the thesis of Lebesgue, and most particularly in the last decade, the study of the problem of Plateau and associated questions has made remarkable advances. This short book is an excellent record of that progress. Chapter I is concerned with surfaces, with Lebesgue area and its ins representation, with compound mappings and wich representations of Jordan curves. Chapter II collects the necessary differential properties of minimal surfaces. In Chapter III the statement of the problem of Plateau, “to find a minimal surface bounded by a given curve”, is analyzed; with restriction to surfaces of the type of the circle, four statements of the parametric problem are distinguished, according to tbe nature of singularities allowed on the surface. The interrelations of these problems are discussed. Properties of minimal surfaces in the large are developed in so far as needed for Iater chapters. Chapter IV proceeds to the non-parametric problem. A minimal surface \(z = z (x, y)\) with prescribed boundary values satisfying a three point condition is found by use of Haar’s theorem of existence and Radö’s theorem of analyticity. Chapter V takes up the parametric problem. It begins with a statement of the results of Garnier, with just enough detail to show why this solution applies only to not-knotted curves (a detailed report being impossxible within the compass of the book). The solutions of the Plateau problem due to Douglas and to Radö are then given in. detail, only a few computations being omitted. Recent simplifications by both authors are utilized. Chapter VI treats the simultaneous problem of finding for a given boundary a minimal surface which at the same time has the least possible area for that boundary. Radö’s solution is given, then that of Douglas with the extension to curves bounding only surfaees of infinite area, and then a third solution by McShane. Then follow theorems interrelating solutions of the problem of Plateau and that of least area. The book coneludes with a brief review of Douglas’ solution of the Plateau problem for surfaces of the type of the Möbius strip.

MSC:
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature