Imbedded minimal surfaces and their Riemannian surfaces. (Eingebettete Minimalflächen und ihre Riemannschen Flächen.) (German) Zbl 1003.53008

After explaining the theoretical background of the relation between the theory of minimal surfaces and complex analysis, in particular, the global theorems of A. Huber [Comment. Math. Helv. 32, 13-72 (1957; Zbl 0080.15001)] and R. Osserman [Ann. Math. 80, 340-364 (1964; Zbl 0134.38502)], the author discusses how a complete minimal surface with finite total curvature may be constructed from geometric assumptions, that is, from assumptions on “how the surface should look like”. The problems that occur in the construction of the Weierstrass data, the underlying Riemann surface, the Gauss map, and the height functional, are addressed: what type of Riemann surfaces carries suitable functions that may serve as a Gauss map, and what effects may arise when trying to solve the period problem.
Many examples are given, that illustrate the discussed methods as well as the occuring problems, and the included figures do not only serve to give the reader a better picture of the surfaces discussed as examples but also illustrate that computer experiments have to be used with caution as a figure of the [nonexisting: see M. Weber, Calc. Var. Partial Differ. Equ. 7, 373-379 (1998; Zbl 1008.53009)] Horgan surface illustrates.
In the reviewer’s opinion the present article, by one of the best specialists in the field, is very well readable and points out relations between many topics that are relevant in minimal surface theory. The article shall be recommended to everyone interested in the construction of minimal surfaces. Even though many topics are touched upon the article should be well suited for students who start to work in the field also, since the author always comes very quickly to the heart of the matter without too much emphasis on technical details.


53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q05 Minimal surfaces and optimization
30F30 Differentials on Riemann surfaces
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)