Bi-eigenmaps and biharmonic submanifolds in a sphere

TL;DR

This paper classifies biharmonic submanifolds and maps in spheres defined by bi-eigenmaps and buckling eigenmaps, extending Takahashi's minimal submanifold characterization to biharmonic cases.

Contribution

It provides a classification of biharmonic submanifolds and maps in spheres based on bi-eigenmaps and buckling eigenmaps, generalizing existing minimal submanifold results.

Findings

Classification of biharmonic submanifolds in spheres.

Classification of biharmonic bi-eigenmaps and buckling eigenmaps.

Extension of Takahashi's minimal submanifold characterization.

Abstract

In this note, we classify biharmonic submanifolds in a sphere defined by bi-eigenmaps ($\Delta^2 \phi=\lambda \phi$) 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.