Liouville Type Theorem for (F, F')p-Harmonic Maps on Foliations

In this paper, we study (F, F')(p)-harmonic maps between foliated Riemannian manifolds (M, g, F) and (M', g', F'). A (F, F')(p)-harmonic map phi : (M, g, F) -> (M', g', F') is a critical point of the transversal p-energy functional E-B,E-p. Trivially, (F, F')(2)-harmonic map is (F, F')-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 F is minimal, then two definitons are equivalent. Firstly, we give the first and second variational formulas for (F, F')(p)-harmonic maps. Next, we investigate the generalized Weitzenbock type formula and the Liouville type theorem for (F, F')(p)-harmonic map.

Automated summary

In this paper, we study (F, F')(p)-harmonic maps between foliated Riemannian manifolds (M, g, F) and (M', g', F'). Trivially, (F, F')(2)-harmonic map is (F, F')-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 F is minimal, then two definitons are equivalent. Firstly, we give the first and second variational formulas for (F, F')(p)-harmonic maps. Next, we investigate the generalized Weitzenbock type formula and the Liouville type theorem for (F, F')(p)-harmonic map.