Stable $f$-harmonic maps on sphere
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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.
Abstract
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.
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