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On embedded complete minimal surfaces of genus zero. (English) Zbl 0719.53004

The authors prove that the Riemann sphere \(S^ 2\) punctured at N points, \(N\geq 3\), cannot be embedded in \({\mathbb{R}}^ 3\) as a complete minimal surface with finite total curvature. This important result together with the works of R. Osserman [Ann. Math., II. Ser. 80, 340-364 (1964; Zbl 0134.385)] and of L. Jorge and W. Meeks [Topology 22, 203-221 (1983; Zbl 0517.53008)] show that the plane and the catenoid are the only embedded complete minimal surfaces of finite total curvature and genus zero in \({\mathbb{R}}^ 3\).

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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