Stability of stationary maps of a functional related to pullbacks of metrics

Let (M,g) and (N,h) be Riemannian manifolds without boundary. We consider the functionalΦ(f)=∫M‖f⁎h‖2dvg for any smooth map f:M→N, where dvgdvg is the volume form on (M,g), and ‖f⁎h‖‖f⁎h‖ denotes the norm of the pullback f⁎hf⁎h of the metric h by the map f  . We study stationary maps for the functional Φ(f)Φ(f), and show that stable stationary maps from or into minimal submanifolds in the unit spheres are rare if Ricci curvatures of submanifolds are large. Symmetric spaces of some type, which are minimally and isometrically immersed in the unit spheres, are treated in detail.