A family of examples of harmonic maps into the sphere with one point singularity

Abstract

The radial map $u\left(x\right)=\frac{x}{\Vert x\Vert }$ is a well-known example of a harmonic map into the spheres with a point singularity at $x=0$. In our previous paper (Misawa and Nakauchi, 2022) we give two examples of harmonic maps into the standard spheres of higher dimension with a singularity of a polynomial of $\frac{x_1}{\Vert x\Vert },\cdots \frac{x_m}{\Vert x\Vert }$ of degree 2 and degree 3 respectively. In Fujioka (2020) uses our arguments to give an example of a harmonic map into the sphere with a singularity of a polynomial of degree 4. In this paper we give a family of examples of harmonic maps with a point singularity of a polynomial of higher degree.