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The Plateau problem. (Das Plateausche Problem.) (German) Zbl 0810.53004
This article is the summary of a lecture delivered by the author at the 1992 annual meeting of the Deutsche Mathematiker-Vereinigung which took place at the Humboldt University, Berlin. It contains a brief sketch of some of the main developments of Plateau’s problem, which is the question, if an arbitrary space curve can span a minimal surface and if so, what can be said about these surfaces. After some remarks about the early history of the problem the author describes the classical existence theorems of Rad√≥-Douglas and Douglas and some of the modern extensions of this theory, mainly relating to regularity and embeddedness of solutions. In the next section the application of methods of global analysis to Plateau’s problem is discussed. This includes statements on the set of minimal surfaces spanned by a generic curve and multiplicity results, in particular a Morse theory for Plateau’s problem. To the latter the author himself has made important contributions. He then goes on by describing the approach to the higher dimensional Plateau problem via geometric measure theory. The article concludes with a few remarks on non-compact minimal surfaces and other boundary conditions. Though the author touches the most important developments in Plateau’s problem, the present article – due to the obvious limitations of a one hour lecture - - is of course not a complete historical account of the problem. For example, the important contributions of R. Courant or Ch. B. Morrey are not mentioned.
MSC:
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53-03 History of differential geometry
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
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