The Plateau problem
type: Item
instance of: scholarly article
title: The Plateau problem
zbMATH Open document ID: 0810.53004
author: Michael Struwe
published in: Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV)
publication date: 1995-04-09T00:00:00Z
review text: 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 Radó-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.
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53-03
Mathematics Subject Classification ID: 53-02
zbMATH DE Number: 617512
zbMATH Keywords: Plateau problem
zbMATH Keywords: regularity
zbMATH Keywords: embeddedness
zbMATH Keywords: Morse theory
zbMATH Keywords: geometric measure theory
zbMATH Keywords: survey
reviewed by: F. Tomi
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 617512