The minimal hypersurfaces in E4 are the only hypersurfaces possessing the following property: Its mean curvature vector field is harmonic. IntroductionLet x: M --t Em be an immersion from an n-dimensional, connected manifold M into the Euclidean rn-space Em. With respect to the Riemannian metric g on M induced from the Euclidean metric of the ambient space Em, M is a Riemannian manifold ( M , g). Denote by A the Laplacian operator of the Riemannian manifold (M, g). Olle of the most important formulas in Differential Geometry of submanifolds is (0.1) Ax = -n H , where fi is the mean curvature vector field of the immersion, and x also denotes the position vector field of M in Em. Formula (0.1) implies that the immersion is minimal (Z? = 0) if and only if the immersion is harmonic, that is, Ax = 0. An isometric immersion x: M -+ Em is called biharmonic if we have (0.2) A2x = 0, that is, A f i = 0 . It is obvious that minimal immersions are biharmonic. In [2] B.-Y. CHEN had asked the following simple geometric question: Other than minimal submanifolds of Em, which submanifolds of Em are biharmonic? Obviously, there are no compact biharmonic submanifolds in Em, as it follows from (0.1)and (0.2). Moreover, I. DIMITRIC in his paper [6] had proved that a biharmonic submanifold M of a Euclidean space is minimal if it is one of the following submanifolds: a curve, a submanifold with constant mean curvature, a hypersurface with a most two distinct principal curvatures or a pseudo-umbilical submanifold of dimension different from 4.

We investigate complete minimal hypersurfaces in the Euclidean space R 4 , with Gauss-Kronecker curvature identically zero. We prove that, if f : M 3 → R 4 is a complete minimal hypersurface with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature bounded from below, then f (M 3 ) splits as a Euclidean product L 2 × R, where L 2 is a complete minimal surface in R 3 with Gaussian curvature bounded from below.2000 Mathematics Subject Classification. 53C42.

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We consider hypersurfaces of E n+l whose position vector x satisfies Ax = Ax + B , where A is the induced Laplacian, and prove that these are open parts of minimal hypersurfaces, hyperspheres or generalized circular cylinders.

Abstract. We investigate 3-dimensional complete minimal hypersurfaces in the hyperbolic space H 4 with Gauss-Kronecker curvature identically zero. More precisely, we give a classification of complete minimal hypersurfaces with GaussKronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature bounded from below.