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Minimal disks and compact hypersurfaces in Euclidean space. (English) Zbl 0574.53038

From the authors’ abstract: ”Let \(M^ n\) be a smooth connected compact hypersurface in \(E^{n+1}\), let \(A^{n+1}\) be the unbounded component of \(E^{n+1}-M^ n\), and let \(\kappa \leq \kappa_ 2\leq...\leq \kappa_ n\) be the principal curvatures of \(M^ n\) with respect to the unit normal pointing into \(A^{n+1}\). It is proven that if \(\kappa_ 2+...+\kappa_ n<0\), then \(A^{n+1}\) is simply connected. Note that if \(n=2\), the hypotheses imply that \(M^ 2\) is convex and \(A^ 3\) is obviously simply connected.”
Reviewer: F.Gackstatter

MSC:

53C40 Global submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
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