Bi-eigenmaps and biharmonic submanifolds in a sphere

Abstract

In this note, we classify biharmonic submanifolds in a sphere defined by bi-eigenmaps (${\Delta }^2\varphi =\lambda \varphi$) or buckling eigenmaps (${\Delta }^2\varphi =-\mu \Delta \varphi$). 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.

8 pages with major improvement of version 2