论文标题
$ l^2_f $谐波1型在平滑度量度量空间上，$λ_1（Δ_F）$
$L^2_f$ harmonic 1-forms on smooth metric measure spaces with positive $λ_1(Δ_f)$
论文作者
论文摘要
在本文中，我们研究了消失的结果，以完全平滑的度量度量空间$（m^n，g，\ \ \ \ mathrm {e}^{ - f} \ mathrm {d} v）$，具有各种负面$ m $ m $ m $ -bakry-émricci-émricci曲率下限，$ $ $ $ $ $ $ up $ up $ up $ up lap lap lap lap lap lap lap lap lap。即$ \ mathrm {ric} _ {m，n} \ geq-aλ_1（δ_f）-b $ for $ 0 <a \ leq \ dfrac {m} {m-1} {m-1}，b \ geq0 $。特别是，我们考虑了三种主要情况，用于不同的$ A $和$ b $，无论有或没有条件，$λ_1（Δ_F）$。这些结果是粪便和vieira的扩展，以及Li-Wang，Dung-Sung和Vieira的加权概括。
In this paper, we study vanishing and splitting results on a complete smooth metric measure space $(M^n,g,\mathrm{e}^{-f}\mathrm{d}v)$ with various negative $m$-Bakry-Émery-Ricci curvature lower bounds in terms of the first spectrum $λ_1(Δ_f)$ of the weighted Laplacian $Δ_f$, i.e. $\mathrm{Ric}_{m,n}\geq -aλ_1(Δ_f)-b$ for $0<a\leq\dfrac{m}{m-1}, b\geq0$. In particular, we consider three main cases for different $a$ and $b$ with or without conditions on $λ_1(Δ_f)$. These results are extensions of Dung and Vieira, and weighted generalizations of Li-Wang, Dung-Sung and Vieira.
