Liouville theorem for $V$-harmonic maps under non-negative $(m,V)$-Ricci curvature for non-positive $m$

Abstract

Let $V$ be a $C^1$-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 $m\in [-\infty ,0]\cup [n,+\infty ]$ into Cartan-Hadam\-ard manifolds, which extends Cheng's Liouville theorem proved 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 $m\in [-\infty ,0]\cup [n,+\infty ]$ into regular geodesic balls of Riemannian manifolds with positive upper sectional curvature bound, which extends the results of Hildebrandt-Jost-Wideman and Choi. Our probabilistic proof of Liouville theorem for several growth $V$-harmonic maps into Hadamard manifolds enhances an incomplete argument by Stafford. Our results extend the results due to Chen-Jost-Qiu\cite{ChenJostQiu} and Qiu\cite{Qiu} in the case of $m=+\infty$ on the Liouville theorem for bounded $V$-harmonic maps from complete Riemannian manifolds with non-negative $(\infty ,V)$-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 ${\Delta }_V$-diffusion processes on manifolds. Our results are new even in the case $V=\nabla f$ for $f\in C^2(M)$.