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Surfaces minimales dans \(R^ 3\). (Minimal surfaces in \({\mathbb{R}}^ 3)\). (French) Zbl 0708.53008

Sémin. Théor. Spectrale Géom. 7, Année 1988-1989, 53-91 (1989).
[For the entire collection see Zbl 0699.00033.]
In this lecture note the author gives an introduction to the theory of minimal surfaces in Euclidean three space. Classical and new examples, the connections to complex analysis and Osserman’s important results are represented. At the end of the paper the well known embedded minimal surface of the author from his thesis in 1982 is constructed.
It is surprising to see that the surface of Enneper type with one handle (see the preceding review) and Costa’s surface can be constructed in a similar way: Take the elliptic functions of Weierstraß belonging to the square lattice and the Weierstraß representation formula for minimal surfaces. With \(f=\wp\) and \(g=a \wp '/\wp,\) \(a=\sqrt{3\pi /2g_ 2},\) we get the Enneper type surface, with \(f=\wp\) and \(g=b/\wp '\), \(b=\sqrt{2g_ 2\pi},\) we come to Costa’s surface. According to a widespread current practice in the theory of minimal surfaces the author does not go to primary sources in some of his citations.
Reviewer: F.Gackstatter

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
33E05 Elliptic functions and integrals
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
Full Text: EuDML