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 $\beta $ from a complete Riemannian manifold with Bakry--\'Emery Ricci curvature bounded below by a constant $-A$ to a Riemannian manifold with sectional curvature bounded above by a negative constant $-B$, then $$ u^*h\le \frac{Ac(\beta )}{BD(\beta )}g \le \frac{A\beta^2}{B}g, $$ where $c(\beta )=\beta^2/(1+\beta^2)$ and $D(\beta )=1-\lfloor 1/c(\beta )\rfloor c(\beta )^2-(1-\lfloor 1/c(\beta )\rfloor c(\beta ))^2$. Equality in the second inequality holds if and only if $\beta =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.