AI 中文总结
该研究建立了$V$-调和映射的精细化Schwarz引理，推导了更紧的映射上界公式与等号成立条件，改进了此前调和映射与$f$-调和映射的相关界，并给出了主定理的应用。
AI 中文摘要
在这篇注记中，我们建立了针对$V$-调和映射的精细化Schwarz引理。具体而言，我们证明：若$u$是从Bakry--Émery Ricci曲率下确界为常数$-A$的完备黎曼流形，映到截面曲率上确界为负常数$-B$的黎曼流形的、具有$β$阶广义伸缩率的$V$-调和映射，则有$$u^*h\le \frac{Ac(β)}{BD(β)}g \le \frac{Aβ^2}{B}g,$$其中$c(β)=β^2/(1+β^2)$，$D(β)=1-\lfloor 1/c(β)\rfloor c(β)^2-(1-\lfloor 1/c(β)\rfloor c(β))^2$。第二个不等式的等号成立当且仅当$β=1,\\ 1/\sqrt{2},\\ 1/\sqrt{3}, \cdots $。我们的结果改进了Shen（《J. Reine Angew. Math.》，1984）针对调和映射、以及Chen--Li--Qiu（《Nonlinear Anal.》，2022）针对$f$-调和映射得到的先前界。我们还给出了主定理的若干应用。
英文摘要
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.
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