A volume decreasing theorem for $$\mathbf{}$$-harmonic maps and applications

We establish a volume decreasing result for V-harmonic maps between Riemannian manifolds. We apply this result to obtain corresponding results for Weyl harmonic maps from conformal Weyl manifolds to Riemannian manifolds. We also obtain corresponding results for holomorphic maps from almost Hermitian manifolds to quasi-Kähler manifolds, which generalize or improve the partial results in Goldberg and Har’El (Bull Soc Math Grèce 18(1):141–148, 1977, J Differ Geom 14(1):67–80, 1979).