Plateau’s problem and the calculus of variations. (English) Zbl 0694.49028

“Given a closed curve \(\Gamma \in {\mathbb{R}}^ 3\) find a surface \(\Sigma\) of least area such that \(\partial \Sigma =\Gamma.''\) This is the classical Plateau problem which was solved in 1930/31 independently by J. Douglas and T. Radó. The book treats this and related geometric variational problems and is divided ito two parts. In part A (The “classical” Plateau problem for disctype minimal surfaces) the first chapter is devoted to the classical solution of the two-dimensional parametric Plateau problem. Struwe presents in detail the existence proof and also describes a number of regularity results such as boundary regularity, existence of branch points, and embeddedness. The second chapter of part A contains a presentation of abstract variational techniques like Ljusternik-Schnirelman theory, the mountain-pass lemma, and Morse theory - all adapted to the theory of minimal surfaces. In particular, a suitable Palais-Smale condition has to be verified which many mathematicians did not expect to hold. These methods - due to the author himself - are then applied to produce unstable minimal surfaces. In a final section the author discusses the classical work of Morse- Tompkins and Shiffman as well as the more recent theory of Böhme and Tromba.
In part B surfaces of constant mean curvature are considered. Here, following mainly Hildebrandt, the author first proves the existence of so called small solutions that one gets under suitable conditions on the mean curvature by using a minimization technique. This chapter also contains Heinz’ non-existence result and a section devoted to regularity questions. In the last chapter unstable H-surfaces are investigated. Here, the author develops a Lusternik-Schnirelman theory and a Morse theory for ‘small’ H-surfaces. However, by looking at simple closed Jordan curves on a sphere one might in general expect two solutions, a small one and a large one. This existence question regarding ‘large’ solutions has been called ‘Rellich’s conjecture’. It was finally answered in 1982 independently by the author with an important contribution by Steffen and by Brezis and Coron. This theory is presented in the last two sections of part B.
The book is written in a more analytic than geometric language which due to the difficulty of the subject makes it a little bit abstract in some places. Nevertheless, by his precise and clear style the author succeeds in making it rather easy for the reader to follow him. Moreover, since there are many historical remarks and since the author describes and explains a number of difficult results and alternative theories due to other authors the reader does not get lost in too many details. Although less than 150 pages the book constains a wealth of material some of which has only been obtained in the last decade by the author himself. This little book is a pleasure to read and I can only recommend it highly both as a textbook and as a research monograph.
Reviewer: M.Grüter


49Q05 Minimal surfaces and optimization
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
49-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control