Liouville Type Theorem for $({\mathcal {F}},{\mathcal {F}}')_{p}$-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 $\phi :(M,g,{\mathcal {F}})\rightarrow (M^{\prime }, g^{\prime },{\mathcal {F}}^{\prime })$ 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 the transversal energy functional $E_B$. 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 $\tau _b(\phi )=0$. 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.