Vanishing results of the F-stress energy tensor

Abstract

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 $\Phi_{S,F}$-energy functional for
maps. The critical map of this functional is called a $\Phi_{S,F}$-harmonic
map. We obtain some vanishing results of $S_F$ by studying Liouville theorems for the $\Phi_{S,F}$-harmonic map. Firstly, we find that the equation of $\Phi_{S,F}$-harmonic map with respect to the metric torsion connection
coincides with that of $\Phi_{S,F}$-harmonic map with respect to the Levi-Civita connection. This shows a rigidity signature of $\Phi_{S,F}$-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 $\Phi_{S,F}$-harmonic map by assuming either growth
condition of the $\Phi_{S,F}$-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 problemson on starlike domains for the $\Phi_{S,F}$-harmonic map. These vanishing theorems extend some results in
[18,19] where $F(t)$ are given as $t$ and $(2t)^{p/2}/p$ $(p\geq 2)$, respectively, and target manifolds are endowed with Levi-Civita connections.