Electron. J. Diff. Eqns., Vol. 2006(2006), No. 15, pp. 1-9.

A Liouville theorem for $F$-harmonic maps with finite $F$-energy

M'hamed Kassi

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 $d_{F}:=\sup(tF'(t)(F(t))^{-1})$ less than  $\infty$. We show that if $d_{F}$ less than $m/2$, 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.

Show me the PDF file (212K), TEX file, and other files for this article.

M'hamed Kassi
Equipe d'Analyse Complexe, Laboratoire d'Analyse Fonctionnelle Harmonique et Complexe
Département de Mathématiques, Faculté des Sciences
Université Ibn Tofail, Kénitra, Maroc
email: mhamedkassi@yahoo.fr

Return to the EJDE web page