Biharmonic homogeneous polynomial maps between spheres

TL;DR

This paper characterizes biharmonic maps between spheres, constructs examples from a torus to a sphere, and classifies quadratic biharmonic maps between spheres, advancing understanding of biharmonic polynomial maps in differential geometry.

Contribution

It provides a characterization formula for biharmonic maps in spheres and classifies quadratic biharmonic maps between spheres, including explicit constructions and classifications.

Findings

Characterization formula for biharmonic maps in Euclidean spheres

Construction of biharmonic maps from a torus to a sphere

Classification of proper biharmonic quadratic forms 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 $\mathbb{S}^n$, $n \geq 2$, from $\mathbb{S}^m$ to $\mathbb{S}^2$, $m \geq 2$, and from $\mathbb{S}^m$ to $\mathbb{S}^3$, $m \geq 2$.