A Schwarz lemma and a Liouville theorem for generalized harmonic maps.

When the sectional curvature of the target manifold is negative, we establish a Schwarz lemma for f -harmonic maps, if the dimension of the domain and the target is large, the result improves Theorem 3 in Chen and Zhao (2017) for the case of V = ∇ f. When the sectional curvature of the target is nonpositive, we obtain a Liouville theorem for the general V -harmonic maps, as a consequence, any V -harmonic function u , satisfying | u (x) | = o (r (x) ) , on a complete Riemannian manifold with nonnegative Bakry–Emery–Ricci curvature is a constant. We also give some applications on gradient Ricci solitons and gradient solitons with potential which are solutions to Ricci-harmonic flow. [ABSTRACT FROM AUTHOR]