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Existence result for minimal hypersurfaces with a prescribed finite number of planar ends. (English) Zbl 0992.53011
By use of tools from nonlinear analysis some authors produced new minimal surfaces in \(R^3\) [e.g., N. Kapouleas, J. Differ. Geom. 47, 95-169 (1997; Zbl 0936.53006)]; some authors studied the properties of the moduli space of such surfaces [e.g., J. Perez and A. Ros, Indiana Univ. Math. J. 45, 177-204 (1996; Zbl 0864.53008)]. Parallel to what has been done for minimal surfaces in \(R^3\), the authors develop a gluing procedure to produce for any \(k\geq 2\) and any \(n\geq 3\) complete immersed minimal hypersurfaces of \(R^{n+1}\) which have \(k\) planar ends. The surfaces are of the topological type of a sphere with \(k\) punctures and they all have finite total curvature.

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53A05 Surfaces in Euclidean and related spaces
47H10 Fixed-point theorems
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