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 eigenvalue \(\lambda _1(\Delta _f)\) of the weighted Laplacian \(\Delta _f\), i.e., \(\mathrm {Ric}_{m,n}\ge -a\lambda _1(\Delta _f)-b\) for \(0<a\le \dfrac{m}{m-1}, b\ge 0\). In particular, we consider three main cases for different a and b with or without conditions on \(\lambda _1(\Delta _f)\). These results are extensions of Dung and Vieira, and weighted generalizations of Li-Wang, Dung-Sung, and Vieira.

在本文中，我们研究消失和在一个完整的平滑度量测量空间拆分结果\（（M ^ N，G，\ mathrm {E} ^ { - F} \ mathrm {d} V）\）与各种负米-根据加权拉普拉斯算子\（\ Delta _f \）的第一特征值\（\ lambda _1（\ Delta _f）\）的Bakry-ÉmeryRicci曲率下界，即\（\ mathrm {Ric} _ {m， n} \ ge -a \ lambda _1（\ Delta _f）-b \）表示\（0 <a \ le \ dfrac {m} {m-1}，b \ ge 0 \）。特别是，我们考虑了不同的a和b在\（\ lambda _1（\ Delta _f）\）上有无条件的三种主要情况。这些结果是Dung和Vieira的扩展，以及Li-Wang，Dung-Sung和Vieira的加权归纳。