Complete $\lambda $-hypersurfaces with constant squared norm of the second fundamental form in the Euclidean space $\mathbb R^{4}$

Abstract

Under the assumption that the quasi-Gauss-Kronecker curvature Kq is identically zero, we give a complete classification of 3-dimensional complete λ-hypersurfaces with constant squared norm S of the second fundamental form in the Euclidean space R 4 , where S = i,j h 2 ij , Kq = det(hij − 1 3 Hδij), with hij being the components of the second fundamental form.