Instability of a family of examples of harmonic maps

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.