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Limits of minimal surfaces and Scherk’s fifth surface. (English) Zbl 0709.53006

The subject of this paper is the convergence of the family of complete minimal surfaces \(M_ k\) embedded in \(R^ 3\) of genus \(k\geq 1\) and three ends. In this family \(M_ 1\) is the surface discovered by the reviewer [Bol. Soc. Bras. Mat. 15, 47-54 (1984; Zbl 0613.53002)] and \(M_ k\), \(k>1\), are the examples constructed by the authors inspired by \(M_ 1\) [Ann. Math., II. Ser. 131, No.1, 1-34 (1990; Zbl 0695.53004)]. The results are the following: i) The \(M_ k\) converge to Scherk’s fifth surface when the Gauss curvature of \(M_ k\) is normalized to have minimum value -1 at the point of these surfaces that lies at the origin of \(R^ 3\). If the \(M_ k\) are normalized to have logarithmic growth 1 on their catenoid ends and the horizontal lines of \(M_ k\) cross at \(\vec O\), then the \(M_ k\) converge to \(C\cup P\), where \(C=\{\cosh x_ 3=| (x_ 1,x_ 2)| \}\) is the standard catenoid with logarithmic growth 1 and P is the \((x_ 1,x_ 2)\) plane.
Reviewer: C.J.Costa

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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