The energy density of biharmonic quadratic maps between spheres

Abstract

In this paper, we first prove that a quadratic form from ${\mathbb{S}}^m$ to ${\mathbb{S}}^n$ is non-harmonic biharmonic if and only if it has constant energy density $(m+1)/2$. Then, we give a positive answer to an open problem concerning the structure of non-harmonic biharmonic quadratic forms. As a direct application, using classification results for harmonic quadratic forms, we infer classification results for non-harmonic biharmonic quadratic forms.