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

TL;DR

This paper proves Liouville theorems for V-harmonic maps under non-negative (m, V)-Ricci curvature, extending previous results and connecting harmonic map properties with diffusion process recurrence on manifolds.

Contribution

It extends Liouville theorems for V-harmonic maps to broader curvature conditions and growth types, introduces probabilistic proofs, and links harmonic map properties with diffusion recurrence.

Findings

Liouville theorem for V-harmonic maps with various growth conditions

Extension of Cheng's Liouville theorem to new curvature regimes

Connection established between Liouville property and diffusion recurrence

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 $f\in C^2(M)$.