Weak conformality of stable stationary maps for a functional related to conformality

Abstract

Let (M,g), (N,h) be compact 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 weakly conformal. The norm ‖Tf‖ is a quantity which is a measure of conformality of f at each point. In Nakauchi (2011) [5] the second author introduced maps which are critical points of the functional Φ(f)=∫M‖Tf‖2dvg. We call such maps C-stationary maps. Any conformal map or more generally any weakly conformal map is a C-stationary map. It is of interest to find when a C-stationary map is a (weakly) conformal map.In this paper we prove the following result. If f is a stable C-stationary map from the standard sphere Sm(m⩾5) or into the standard sphere Sn(n⩾5), then f is a weakly conformal map.