Instability of a family of examples of harmonic maps

Abstract

The radial map u(x) $=$ $\frac{x}{\Vert x\Vert }$ is a well-known example of a harmonic map from ${\mathbb {R}}^m\,-\,\{0\}$ into the spheres ${\mathbb {S}}^{m-1}$ with a point singularity at x $=$ 0. In Nakauchi (Examples Counterexamples 3:100107, 2023), the author constructed recursively a family of harmonic maps $u^{(n)}$ into ${\mathbb {S}}^{m^n-1}$ with a point singularity at the origin $(n = 1,\,2,\ldots )$, such that $u^{(1)}$ is the above radial map. It is known that for m $\ge$ 3, the radial map $u^{(1)}$ is not only stable as a harmonic map but also a minimizer of the energy of harmonic maps. In this paper, we show that for n $\ge$ 2, $u^{(n)}$ may be unstable as a harmonic map. Indeed we prove that under the assumption n > ${\displaystyle \frac{\sqrt{3}-1}{2}\,(m-1)}$ $(m \ge 3$, $n \ge 2)$, the map $u^{(n)}$ is unstable as a harmonic map. It is remarkable that they are unstable and our result gives many examples of unstable harmonic maps into the spheres with a point singularity at the origin.