Hasanis, Th.; Savas-Halilaj, A.; Vlachos, Th. Complete minimal hypersurfaces in \(\mathbb{S}^4\) with zero Gauss-Kronecker curvature. (English) Zbl 1114.53056 Math. Proc. Camb. Philos. Soc. 142, No. 1, 125-132 (2007). The main theorem of this paper is as follows. Let \(M^3\) be a 3-dimensional complete Riemannian manifold and \(f: M^3\to\mathbb{S}^4\) a minimal isometric immersion with Gauss-Kronecker curvature identically zero. If the square \(S\) of the second fundamental form is nowhere zero and bounded from above, then \(f(M^3)\) is the image of the polar map associated with a superminimal immersion \(g: M^2\to\mathbb{S}^4\) with positive normal curvature. Moreover, if \(S\) is bounded away from zero, then \(M^2\) is diffeomorphic to the sphere \(\mathbb{S}^2\) or to the projective plane \(\mathbb{R}\mathbb{P}^2\) and \(f(M^3)\) is compact. Reviewer: Ülo Lumiste (Tartu) Cited in 5 Documents MSC: 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) Keywords:four-dimensional sphere; minimal hypersurface; second fundamental form; Gauss-Kronecker curvature PDFBibTeX XMLCite \textit{Th. Hasanis} et al., Math. Proc. Camb. Philos. Soc. 142, No. 1, 125--132 (2007; Zbl 1114.53056) Full Text: DOI arXiv