Biharmonic homogeneous polynomial maps between spheres

Abstract

In this paper we first prove a characterization formula for biharmonic maps in Euclidean spheres and, as an application, we construct a family of biharmonic maps from a flat $2$-dimensional torus $\mathbb{T}$ into the $3$-dimensional unit Euclidean sphere ${\mathbb{S}}^3$. Then, for the special case of maps between spheres whose components are given by homogeneous polynomials of the same degree, we find a more specific form for their bitension field. Further, we apply this formula to the case when the degree is $2$, and we obtain the classification of all proper biharmonic quadratic forms from ${\mathbb{S}}^1$ to ${\mathbb{S}}^n$, $n\ge 2$, from ${\mathbb{S}}^m$ to ${\mathbb{S}}^2$, $m\ge 2$, and from ${\mathbb{S}}^m$ to ${\mathbb{S}}^3$, $m\ge 2$.

36 pages