A Schwarz lemma and a Liouville theorem for generalized harmonic maps

Abstract

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.