Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 15, pp. 1-9.

Title: A Liouville theorem for $F$-harmonic maps 
       with finite $F$-energy
       
Author: M'hamed Kassi (Univ. Ibn Tofail, Kenitra, Maroc)

Abstract: 
 Let $(M,g)$ be a $m$-dimensional complete Riemannian manifold with a
 pole, and $(N,h)$ a Riemannian manifold. Let
 $F : \mathbb{R}^{+}\to \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 : M\to N$ with finite  $F$-energy (i.e a local extremal of
 $E_{F}(u):= \int_{M} F(\vert du\vert^{2}/2)dV_{g}$ and $E_{F}(u)$
 is finite) is a constant map provided that the radial curvature of
 $M$ satisfies  a pinching condition depending to $d_{F}$.
 
Submitted March 24, 2005. Published January 31, 2006.
Math Subject Classifications: 58E20, 53C21, 58J05.
Key Words: $F$-harmonic maps; Liouville propriety; Stokes formula;
           comparison theorem.