Das Existenzproblem für Minimalflächen. (The existence problem for minimal surfaces). (German) Zbl 0665.53007

In this survey article the author reviews the most important concepts which have been used in proving existence theorems for minimal submanifolds. Special emphasis is given to surfaces (i.e. 2-dimensional submanifolds) and boundary value problems such as the famous problem of Plateau. After a historical introduction the author sketches Courant’s version of the classical Douglas-Radò solution to Plateau’s problem for disc type minimal surfaces and indicates various developments initiated by this key result. Some of these developments are reviewed in the subsequent paragraphs: Plateau’s problem for higher genus surfaces, Morse theory for minimal surfaces, Geometric Measure Theory and the higher dimensional Plateau problem, embedded minimal surfaces, saddle point constructions and the concept of varifolds. The paper concludes with a sketch of a physically more realistic mathematical model for soap films introduced by Almgren, and with a discussion of uniqueness questions. This paper is a very readable introduction to the subject and leads the reader very quickly to the frontier of current research.
Reviewer: F.Tomi


53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q05 Minimal surfaces and optimization
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry