Bi-eigenmaps and biharmonic submanifolds in a sphere

In this note, we classify biharmonic submanifolds in a sphere defined by bi-eigenmaps (${\mathrm{\Delta }}^2\varphi =\lambda \varphi$) or buckling eigenmaps ($\Delta^2 \phi=-\mu \Delta \phi$). We then classify biharmonic bi-eigenmaps and buckling eigenmaps into spheres with constant energy density. The results can be viewed as generalizations of Takahashi’s characterization of minimal submanifolds in a sphere by eigenmaps.