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A general construction, for complete embedded minimal surfaces of finite total curvature in Euclidean three-space, is carried out. In particular, examples with an arbitrary number of ends are given for the first time. The construction amounts to desingularizing the circles of intersection of a collection of coaxial catenoids and planes. The desingularization process uses Scherk's singly periodic surfaces for an approximate construction which is subsequently corrected by singular perturbation methods.
Nikolaos Kapouleas (Wed,) studied this question.