𝑉-harmonic morphisms between Riemannian manifolds

A V V -harmonic morphism u : M → N u:M\to N between Riemannian manifolds is a smooth map which pulls back germs of harmonic functions on N N to germs of V V -harmonic functions on M M , where V V is a smooth vector field on M M . In this paper, we give some characterizations and examples of V V -harmonic morphisms. In addition, a dilation estimate and a Liouville-type theorem of V V -harmonic morphisms from noncompact complete manifolds are also established. As applications, we obtain the Liouville-type theorems for V V -harmonic morphisms from complete manifolds of nonnegative Bakry-Émery Ricci curvature, especially complete steady or shrinking Ricci solitons, to manifolds of dimension at least three or compact Riemann surface of genus at least two.