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A variational approach to the existence of complete embedded minimal surfaces. (English) Zbl 0676.53006

In several papers the authors have constructed properly embedded minimal surfaces of finite total curvature in \({\mathbb{R}}^ 3\). The theory of elliptic functions and other analytic methods are the basis of these constructions. In spite of the power of these analytic techniques, they have not been successful in answering qualitative questions about surfaces with a large number of ends. The present paper is an attempt to address such questions and, at the same time, to develop a general variational approach to proving the existence of embedded minimal surfaces. There is an alternative construction of the well known surfaces \(M_ k\) of genus k with three ends.
Reviewer: F.Gackstatter

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49J15 Existence theories for optimal control problems involving ordinary differential equations
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