Liouville theorem for V-harmonic maps under non-negative (m,V)-Ricci curvature for non-positive m
This is a joint work with Professors, Xiang-Dong Li (CAS AMSS), Songzi Li (Renmin University) and Yohei Sakurai (Saitama University). Let V be a C1-vector field on an n-dimensional complete Riemannian manifold (M,g). We prove a Liouville theorem for V-harmonic maps satisfying various growth conditions from complete Riemannian manifolds with non-negative (m,V)-Ricci curvature for m∈[−∞,0]∪[n,+∞] into Cartan-Hadamard manifolds, which extends Cheng's Liouville theorem proved in S.Y. Cheng for sublinear growth harmonic maps from complete Riemannian manifolds with non-negative Ricci curvature into Cartan-Hadamard manifolds. We also prove a Liouville theorem for V-harmonic maps from complete Riemannian manifolds with non-negative (m,V)-Ricci curvature for m∈[−∞,0]∪[n,+∞] into regular geodesic balls of Riemannian manifolds with positive upper sectional curvature bound, which extends the results of Hildebrandt-Jost-Wideman and Choi. Our stochastic proof of Liouville theorem for several growth V-harmonic maps into Hadamard manifolds fills an incomplete argument in \cite{Staff:Liouville}. Our results extend the results due to Chen-Jost-Qiu and Qiu in the case of m=+∞ on the Liouville theorem for bounded V-harmonic maps from complete Riemannian manifolds with non-negative (∞,V)-Ricci curvature into regular geodesic balls of Riemannian manifolds with positive sectional curvature upper bound. Finally, we establish a connection between the Liouville property of V-harmonic maps and the recurrence property of ΔV-diffusion processes on manifolds. Our results are new even in the case V=∇f for f∈C2(M).