Vanishing results of the F-stress energy tensor

In this paper, we study some vanishing results of the F-stress energy tensor S_F associated to the F-energy where the target manifold is equipped with a metric connection having non-vanishing torsion. By estimating the norm of S_F , we introduce a ΦS_F -energy functional for maps. The critical map of this functional is called a ΦS_F -harmonic map. We obtain some vanishing results of S_F by studying Liouville theorems for the ΦS_F -harmonic map. Firstly, we find that the equation of ΦS_F -harmonic map with respect to the metric torsion connection coincides with that of ΦS_F -harmonic map with respect to the Levi-Civita connection. This shows a rigidity signature of ΦS_F -harmonic map being invariant under connection transforms from the Levi-Civita connection to the metric torsion connection. Then, under suitable conditions on the Hessian of the distance function and the degree of F(t) , we derive several Liouville theorems for the ΦS_F -harmonic map by assuming either growth condition of the ΦS_F -energy or an asymptotic condition at the infinity for the maps. In the end of paper, we also obtain the unique constant solution of the constant Dirichlet boundary value problem on starlike domains for the ΦS_F -harmonic map. These vanishing theorems extend some results in [18, 19] where F(t) are given as t and (2t)p/2/p ( p ≥ 2 ), respectively, and target manifolds are endowed with Levi-Civita connections.