A refined Schwarz lemma for $V$-harmonic maps

Abstract

In this note, we establish a refined Schwarz lemma for $V$-harmonic maps. Specifically, we prove that if $u$ is a $V$-harmonic map of generalized dilatation of order $β$ from a complete Riemannian manifold with Bakry--Émery Ricci curvature bounded below by a constant $-A$ to a Riemannian manifold with sectional curvature bounded above by a negative constant $-B$, then $$ u^*h\le \frac{Ac(β)}{BD(β)}g \le \frac{Aβ^2}{B}g, $$ where $c(β)=β^2/(1+β^2)$ and $D(β)=1-\lfloor 1/c(β)\rfloor c(β)^2-(1-\lfloor 1/c(β)\rfloor c(β))^2$. Equality in the second inequality holds if and only if $β=1,\ 1/\sqrt{2},\ 1/\sqrt{3}, \cdots $. Our result improves the previous bounds obtained by Shen (J. Reine Angew. Math., 1984) for harmonic maps and by Chen--Li--Qiu (Nonlinear Anal., 2022) for $f$-harmonic maps. We also present some applications of our main theorem.