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.

The main purpose of this paper is to complete the work initiated by Sbrana in 1909 giving a complete local classification of the nonflat infinitesimally bendable hypersurfaces in Euclidean space.

We introduce a class of minimal submanifolds M n , n ≥ 3, in spheres S n+2 that are ruled by totally geodesic spheres of dimension n − 2. If simply-connected, such a submanifold admits a one-parameter associated family of equally ruled minimal isometric deformations that are genuine. As for compact examples, there are plenty of them but only for dimensions n = 3 and n = 4. In the first case, we have that M 3 must be a S 1 -bundle over a minimal torus T 2 in S 5 and in the second case M 4 has to be a S 2 -bundle over a minimal sphere S 2 in S 6 . In addition, we provide new examples in relation to the well-known Chern-do Carmo-Kobayashi problem since taking the torus T 2 to be flat yields a minimal submanifolds M 3 in S 5 with constant scalar curvature.In several directions, this paper should be considered as a continuation of our work in [8] where a new class of minimal ruled submanifolds M n of Euclidean space R n+2 , n ≥ 3, were studied. These submanifolds lay in codimension two and may be metrically complete regardless the dimension. The rulings are of codimension two in the manifold whereas the rank, that is, the complement of the index of relative nullity, is ρ = 4 (unless n = 3 = ρ) along an open dense subset. If simply-connected, the submanifolds admit a S 1 -parameter family of genuine isometric deformations. Hence, this class of examples should be seen as a new addition to the possible, local or global, classification of Euclidean submanifolds in codimension two that admit genuine isometric deformations; see [8] for a discussion of that open problem.In this paper, we consider a similar construction but for the round sphere as ambient space. We obtain minimal submanifolds M n in S n+2 , n ≥ 3, with similar properties as the ones in the Euclidean space. Notice that being ruled now means that the submanifold carries a foliation by (open subsets of) totally geodesic spheres in S n+2 of dimension n−2. If the manifold is simply-connected, by taking the cones in R n+3 of the components in the associated family in S n+2 we obtain a new class of genuinely deformable Euclidean submanifolds in codimension two but, of course, these are not complete.

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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.