$L_k$-BIHARMONIC HYPERSURFACES IN THE EUCLIDEAN SPACE

Abstract

Chen conjecture states that every Euclidean biharmonic submanifold is minimal. In this paper we consider the Chen conjecture for $ L_k $-operators. The new conjecture ($ L_k $-conjecture) is formulated as follows: If $ L_k^{2}x=0 $ then $ H_{k+1}=0$ where $ x:M^{n}\rightarrow\Bbb{R}^{n+1} $ is an isometric immersion of a Riemannian manifold $ M^n $ into the Euclidean space $ \Bbb{R}^{n+1} $, $ H_{k+1} $ is the $ (k+1) $-th mean curvature of $ M $, and $ L_k $ is the linearized operator of the $ (k+1) $-th mean curvature of the Euclidean hypersurface $ M $. We prove the $ L_k $-conjecture for the hypersurface $ M $ with at most two principal curvatures.