On the equivariant stability of harmonic self-maps of cohomogeneity one manifolds

The systematic study of harmonic self-maps on cohomogeneity one manifolds has recently been initiated by Püttmann and the second named author in [19]. In this article we investigate the corresponding Jacobi equation describing the equivariant stability of such harmonic self-maps. Besides several general statements concerning their equivariant stability we explicitly solve the Jacobi equation for some harmonic self-maps in the cases of spheres, special orthogonal groups and SU(3). In particular, we show by an explicit calculation that for specific cohomogeneity one actions on the sphere the identity map is equivariantly stable.