On some Lr-biharmonic Euclidean hypersurfaces

In decade eighty, Bang-Yen Chen introduced the concept of biharmonic hypersurface in the Euclidean space. An isometrically immersed hypersurface $x : M^{n} \rightarrow \mathbb{E}^{n+1}$ is said to be biharmonic if $\Delta^{2}x = 0$, where $\Delta$ is the Laplace operator. We study the $L_{r}$-biharmonic hypersurfaces as a generalization of biharmonic ones, where Lr is the linearized operator of the $(r + 1)$th mean curvature of the hypersurface and in special case we have $L_{0} = \Delta$. We prove that $L_{r}$-biharmonic hypersurface of $L_{r}$-finite type and also $L_{r}$-biharmonic hypersurface with at most two distinct principal curvatures in Euclidean spaces are r-minimal.