Conformality of rotationally symmetric maps

Abstract

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.Any conformal map or more generally any weakly conformal one is a C-stationary map. However, there exists a C-stationary map which is not a weakly conformal one. (See Example in [3], p. 156.) In the case that the source manifold or the target one is the n-dimensional standard sphere (n≥5), any stable C-stationary map is a weakly conformal one. (See Theorem 1 and 2 in [3].)In this paper we are concerned with rotationally symmetric maps. We prove that any rotationally symmetric smooth map between 4-dimensional model spaces is a C-stationary map if and only if it is a conformal one.