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

TL;DR

This paper investigates $(,')_p$-harmonic maps between foliated Riemannian manifolds, deriving variational formulas, a Weitzenb"ock formula, and establishing a Liouville theorem under certain conditions.

Contribution

It introduces the first and second variational formulas for $(,')_p$-harmonic maps and proves a Liouville type theorem, extending harmonic map theory to foliated manifolds.

Findings

Derived variational formulas for $(,')_p$-harmonic maps.

Established a generalized Weitzenb"ock formula.

Proved a Liouville type theorem for these maps.

Abstract

In this paper, we study $(\mathcal F,\mathcal F')_{p}$-harmonic maps between foliated Riemannian manifolds $(M,g,\mathcal F)$ and $(M',g',\mathcal F')$. A $(\mathcal F,\mathcal F')_{p}$-harmonic map $\phi:(M,g,\mathcal F)\to (M', g',\mathcal F')$ is a critical point of the transversal $p$-energy functional $E_{B,p}$. Trivially, $(\mathcal F,\mathcal F')_2$-harmonic map is $(\mathcal F,\mathcal F')$-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 $\tau_b(\phi)=0$. 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 $(\mathcal F,\mathcal F')_{p}$-harmonic maps. Next, we investigate the generalized Weitzenb\"ock type formula and the Liouville type theorem for $(\mathcal F,\mathcal F')_{p}$-harmonic map.