Conformality of rotationally symmetric maps

Let (M,g), (N,h) be Riemannian manifolds without boundary, and let f be a smooth map from M into N. We consider a covariant symmetric tensor Tf = f⁎h−1m‖df‖2g, where f⁎h denotes the pullback of the metric h by f, and m is the dimension of the manifold M. The tensor Tf vanishes if and only if the map f is a weakly conformal map. The norm ‖Tf‖ is a quantity which is a measure of the conformality of f at each point. In [4] the author introduced maps which are critical points of the functional Econf(f) = ∫M‖Tf‖2dvg in the case that M is compact. We call such maps C-stationary maps. In the case that M is non-compact, f is defined to be a C-stationary map if it is C-stationary on any compact subset of M.