Instability of a family of examples of harmonic maps

The radial map u(x) = x/& Vert;x & Vert; is a well-known example of a harmonic map from R-m-{0} into the spheres Sm-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 Smn-1 with a point singularity at the origin (n=1,2,& mldr;), such that u((1)) is the above radial map. It is known that for m >= 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 >= 2, u((n)) may be unstable as a harmonic map. Indeed we prove that under the assumption n > root 3-1/2(m-1) (m >= 3, n >= 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.