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 $T_f$ $=$ ${\displaystyle f^*h - \frac{1}{m} |df|^2 g}$, where $f^*h$ denotes the pull-back metric of $h$ by $f$. The tensor $T_f$ vanishes if and only if the map $f$ is weakly conformal. The norm $|T_f|$ is a quantity which is a measure of conformality of $f$ at each point. We are concerned with maps which are critical points of the functional $\Phi (f)$ $=$ ${\displaystyle \int_M |T_f|^2dv_g}$. We call such maps {\it 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 maps from the standard sphere ${\mathrm S}^m$ $(m \geq 5)$ or into the standard sphere ${\mathrm S}^n$ $(n \geq 5)$, then $f$ is a weakly conformal map.