A Liouville theorem for F-harmonic maps with finite F-energy
M'hamed Kassi · DOAJ (DOAJ: Directory of Open Access Journals) · 2006
Let $(M,g)$ be a $m$-dimensional complete Riemannian manifold with a pole, and $(N,h)$ a Riemannian manifold. Let $F : mathbb{R}^{+}o mathbb{R}^{+} $ be a strictly increasing $C^{2}$ function such that $F(0)=0$ and $d_{F}:=sup(tF'(t)(F(t))^{-1}) < infty$. We show that if $d_{F} < m/2$, then every $F$-harmonic map $ u : Mo N$ with finite $F$-energy (i.e a local extremal of $E_{F}(u):= int_{M} F(vert duvert^{2}/2)dV_{g}$ and $E_{F}(u)$ is finite) is a constant map provided that the radial curvature of $M$ satisfies a pinching condition depending to $d_{F}$.