Communications of the
Korean Mathematical Society CKMS

In this paper, we prove that any stable $f$-harmonic map $\psi$ from $\mathbb{S}^2$ to $N$ is a holomorphic or anti-holomorphic map, where $N$ is a K\"ahlerian manifold with non-positive holomorphic bisectional curvature and $f$ is a smooth positive function on the sphere $\mathbb{S}^2$ with $\operatorname{Hess}f\leq0$. We also prove that any stable $f$-harmonic map $\psi$ from sphere $\mathbb{S}^n$ $(n>2)$ to Riemannian manifold $N$ is constant.

Keywords: $f$-harmonic maps, $F$-harmonic maps, K\"ahlerian manifold, Bisectional curvature

MSC numbers: 53C43, 58E20