Bulletin of the
Korean Mathematical Society BKMS

We establish a monotonicity formula of $V$-harmonic maps by using the stress-energy tensor. Use the monotonicity formula, we can derive a Liouville type theorem for $V$-harmonic maps. As applications, we also obtain monotonicity and constancy of Weyl harmonic maps from conformal manifolds to Riemannian manifolds and $\pm $holomorphic maps between almost Hermitian manifolds. Finally, a constant boundary-value problem of $V$-harmonic maps is considered.

Keywords: $V$-harmonic map, monotonicity formula, Liouville type theorem, holomorphic map, constant boundary-value problem

MSC numbers: Primary 58E20, 53C43, 35B53, 53C55