Liouville-type results for stationary maps of a class of functional related to pullback metrics

We study a generalized functional related to the pullback metrics (3). We derive the first variation formula which yield stationary maps. We introduce the stress–energy tensor which is naturally linked to conservation law and yield the monotonicity formula via the coarea formula and the comparison theorem in Riemannian geometry. A version of this monotonicity inequalities enables us to derive some Liouville type results. Also, we investigate the constant Dirichlet boundary value problems and the generalized Chern type results for tension field equation with respect to this functional.