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چکیده
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A famous conjecture of Bang-Yen Chen states that every Euclidean biharmonic submanifold is minimal. The conjecture is followed by many researchers ([7, 8, 9, 11]). An isometrically immersed hypersurface x : M^n--->E^{n+1} is called biharmonic if Delta^2 x=0, where Delta is the Laplace operator. In this paper, we consider an advanced version of chen conjecture. we study the L_k-biharmonic spacelike hypersurfaces of L^{n+1}, where L_k is the linearized operator of the first variation of (k+1)-th mean curvature of M^n which is the Laplace operator \Delta in the special case k=0 A spacelike hypersurface x: M^n---->L^{n+1} is called L_k-biharmonic if L_k^2x=0. On any L_k-biharmonic spacelike hypersurfaces x : M^n---->L^{n+1} with at least two distinct principal curvatures, having assumed the k-th mean curvature H_k be constant we will provide that H_{k+1} is constant also. Furthermore, we show that any L_k-biharmonic spacelike hypersurface in L^{n+1} with constant H_k is k-maximal. As the matter of fact, from the point of view of finite type theory, we show that if $M^n$ be of L_k-finite type or it has at most two distinct principal curvatures, then M^n will be a $k$-maximal .
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