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We study a generalized functional related to the pullback metrics (3). Wederive the first variation formula which yield stationary maps. We introducethe stress-energy tensor which is naturally linked to conservation law andyield monotonicity formula via the coarea formula and comparison theorem inRiemannian geometry. A version of this monotonicity inequalities enables us toderive some Liouville type results. Also we investigate the constant Dirichletboundary value problems and the generalized Chern type results for tensionfield equation with respect to this functional.

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