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Some recent developments in the theory of properly embedded minimal surfaces in \(\mathbb{R}^ 3\). (English) Zbl 0789.53003
Séminaire Bourbaki, Vol. 1991/92. Exposés 745-759 (avec table par noms d’auteurs de 1948/49 à 1991/92). Paris: Société Mathématique de France, Astérisque. 206, 463-535 (Exp. No. 759) (1992).
The author gives a survey of the recent work concerning properly embedded minimal surfaces in \(\mathbb{R}^ 3\). The main contents are as follows: The characterizations of the catenoid by R. Schoen and Lopez-Ros, the curvature estimates of stable minimal surfaces, initiated by Heinz for graphs and in general by R. Schoen, the annular end theorem and the strong halfspace theorem of Hoffmann-Meeks, the work of Meeks and the author on periodic minimal surfaces. The main result of the last is that finite topology of the quotient surface implies finite total curvature of this quotient surface. Some applications of this theorem are given. For example, the author shows that the plane and the helicoid are the only simply connected properly embedded minimal surfaces in \(\mathbb{R}^ 3\) with an infinite symmetry group. He also discusses the sum of minimal surfaces and its applications, and finally gives some problems, conjectures and related results.
For the entire collection see [Zbl 0772.00016].

MSC:
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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