Proper Lk-biharmonic Hypersurfaces in The Euclidean Sphere with Two Principal Curvatures

In this paper we classify proper L_k-biharmonic hypersurfaces M , in the unit Euclidean sphere which has two principal curvatures and we show that they are open pieces of standard products of spheres. Also we study proper L_k-biharmonic compact hypersurfaces M with respect to tr(S^۲\circ P_k) and H_k where S is the shape operator, P_k is the Newton transformation and H_k is the k -th mean curvature of M , and by definiteness's assumption of P_k , we show that H_{k+۱} is constant.

Keywords:

L_k operator , biharmonic hypersurfaces , Chen conjecture