We study biharmonic submanifolds of the Euclidean sphere that satisfy certain geometric properties. We classify: (i) the biharmonic hypersur- faces with at most two distinct principal curvatures; (ii) the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudo- umbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curvature vector field. We also study the type, in the sense of B-Y. Chen, of compact proper biharmonic submanifolds with constant mean curvature in spheres

We investigate proper biharmonic hypersurfaces with at most three distinct principal curvatures in space forms. We obtain the full classification of proper biharmonic hypersurfaces in 4-dimensional space forms

We obtain several rigidity results for biharmonic submanifolds in S n with parallel normalized mean curvature vector field. We classify biharmonic submanifolds in S n with parallel normalized mean curvature vector field and with at most two distinct principal curvatures. In particular, we determine all biharmonic surfaces with parallel normalized mean curvature vector field in S n .Then we investigate, for (not necessarily compact) proper biharmonic submanifolds in S n , their type in the sense of B-Y. Chen. We prove: (i) a proper biharmonic submanifold in S n is of 1-type or 2-type if and only if it has constant mean curvature f = 1 or f ∈ (0, 1), respectively; (ii) there are no proper biharmonic 3-type submanifolds with parallel normalized mean curvature vector field in S n .where τ (ϕ) = trace ∇dϕ is the tension field. The functional E 2 is called the bienergy functional. In the particular case when ϕ : (M, g) → (N, h) is a Riemannian immersion, the tension field has the expression τ (ϕ) = mH and equation (1.1) is equivalent to ϕ being a critical point of E 2 . Obviously, any minimal submanifold (H = 0) is biharmonic. The non-harmonic biharmonic submanifolds are called proper biharmonic.The study of proper biharmonic submanifolds is nowadays becoming a very active subject and its popularity initiated with the challenging conjecture of B-Y. Chen: any biharmonic submanifold in an Euclidean space is minimal.

Biharmonic maps between warped products are studied. The main results are:\ud
(i) the condition for the biharmonicity of the inclusion of a Riemannian manifold N into the warped product M ×f 2 theprojectionπ : M×f2 N → M;\ud
N and of\ud
￼(ii) the construction of two new classes of non-harmonic biharmonic maps using products of harmonic maps φ = 1M × ψ : M × N → M × N and warping the metric on their domain or codomain;\ud
(iii) the study of three classes of axially symmetric biharmonic maps, using the warped product setting

We present some results on the boundedness of the mean curvature of proper biharmonic submanifolds in spheres. A partial classification result for proper biharmonic submanifolds with parallel mean curvature vector field in spheres is obtained. Then, we completely classify the proper biharmonic submanifolds in spheres with parallel mean curvature vector field and parallel Weingarten operator associated to the mean curvature vector field.2010 Mathematics Subject Classification. 58E20.

Abstract. In the present paper we survey the most recent classification results for proper biharmonic submanifolds in unit Euclidean spheres. We also obtain some new results concerning geometric properties of proper biharmonic constant mean curvature submanifolds in spheres.

Abstract. In this paper we prove that a constant mean curvature surface is proper-biharmonic in the unit Euclidean sphere S 4 if and only if it is minimal in a hypersphere S 3 (1/ √ 2).
IntroductionBiharmonic maps ϕ : (M, g) → (N, h) between Riemannian manifolds are critical points of the bienergy functionalwhere τ (ϕ) = trace ∇ dϕ is the tension field of ϕ that vanishes for harmonic maps (see [10]). The Euler-Lagrange equation corresponding to E 2 is given by the vanishing of the bitension fieldwhere J ϕ is formally the Jacobi operator of ϕ (see [14]). The operator J ϕ is linear, thus any harmonic map is biharmonic. We call proper-biharmonic the non-harmonic biharmonic maps. The study of proper-biharmonic submanifolds, i.e. submanifolds such that the inclusion map is non-harmonic (non-minimal) biharmonic, constitutes an important research direction in the theory of biharmonic maps. The first ambient spaces considered for their properbiharmonic submanifolds were the spaces of constant sectional curvature. Non-existence results were obtained for proper-biharmonic submanifolds in Euclidean and hyperbolic spaces (see [1, 4, 7, 9, 12]).The case of the Euclidean sphere is different. Indeed, the hypersphere S n−1 (1/ √ 2) and the generalized Clifford torus (1/ √ 2), n 1 + n 2 = n − 1, n 1 = n 2 .2000 Mathematics Subject Classification: Primary 58E20.

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