A Schwarz lemma and a Liouville theorem for generalized harmonic maps. (January 2022) - Chen, Qun; Li, Kaipeng; Qiu, Hongbing
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.