Liouville type theorem for (F;F')p-harmonic maps on foliations

Abstract

In this paper, we study $(\mathcal{F},{\mathcal{F}}^{\prime }{)}_p$-harmonic maps between foliated Riemannian manifolds $(M,g,\mathcal{F})$ and $(M^{\prime },g^{\prime },{\mathcal{F}}^{\prime })$. A $(\mathcal{F},{\mathcal{F}}^{\prime }{)}_p$-harmonic map $\varphi :(M,g,\mathcal{F})\to (M^{\prime },g^{\prime },{\mathcal{F}}^{\prime })$ is a critical point of the transversal $p$-energy functional $E_{B,p}$. Trivially, $(\mathcal{F},{\mathcal{F}}^{\prime }{)}_2$-harmonic map is $(\mathcal{F},{\mathcal{F}}^{\prime })$-harmonic map, which is a critical point of $E_B$. There is another definition of a harmonic map on foliated Riemannian manifolds, called transversally harmonic map, which is a solution of the Euler-Largrange equation ${\tau }_b(\varphi )=0$. Two definitions are not equivalent, but if $\mathcal{F}$ is minimal, then two definitons are equivalent. Firstly, we give the first and second variational formulas for $(\mathcal{F},{\mathcal{F}}^{\prime }{)}_p$-harmonic maps. Next, we investigate the generalized Weitzenböck type formula and the Liouville type theorem for $(\mathcal{F},{\mathcal{F}}^{\prime }{)}_p$-harmonic map.

17pages. This article is updated because the previous version has so many overlaps