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.