Some Results On F - Harmonic Mappings With Potential

In this paper, we introduce the F-harmonic maps with potential H,the (F,H)-energy-density and the (F,H)-stress-energy tensor. We use the(F,H)-stress-energy tensor and the F-stress-energy tensor to obtain the monotonicity formula and vanishing theorems for F-harmonic maps with potential H under some con-ditions on H and F. Some inequalities obtained in the process of proving the monotonicity formulae will be used to investigate the constant Dirichlet bound-ary value problem for F-harmonic maps with potential H.Then we study the stability of F-harmonic maps with potential H.We prove that under some condi-tions on H, F and the principal curvatures of a compact convex hypersurface M in Rm+1,there is no nonconstant stable F-harmonic map with potential H between Mand any compact Riemannian manifold.