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Complete minimal hypersurfaces in \(\mathbb{S}^4\) with zero Gauss-Kronecker curvature. (English) Zbl 1114.53056

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.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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