Communications of the
Korean Mathematical Society CKMS

In this paper, we prove that any stable $f$-harmonic map from sphere $\mathbb{S}^n$ to Riemannian manifold $(N,h)$ is constant, where $f$ is a smooth positive function on $\mathbb{S}^n\times N$ satisfying one condition with $n>2$. We also prove that any stable $f$-harmonic map $\varphi$ from a compact Riemannian manifold $(M,g)$ to $\mathbb{S}^n$ $(n>2)$ is constant where, in this case, $f$ is a smooth positive function on $M\times\mathbb{S}^n$ satisfying $\Delta^{\mathbb{S}^{n}}(f)\circ\varphi\leq 0$.

Keywords: harmonic maps, $f$-harmonic maps, stable $f$-harmonic maps

MSC numbers: Primary 53C43, 58E20