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

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 $d_{F}:=\sup(tF'(t)(F(t))^{-1})$ 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.

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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

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