$L_f^2$ harmonic 1-forms on smooth metric measure spaces with positive ${\lambda }_1({\mathrm{\Delta }}_f)$

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-'Emery-Ricci curvature lower bounds in terms of the first spectrum ${\lambda }_1({\mathrm{\Delta }}_f)$ of the weighted Laplacian ${\mathrm{\Delta }}_f$, i.e. ${\mathrm{R}\mathrm{i}\mathrm{c}}_{m,n}\ge -a{\lambda }_1({\mathrm{\Delta }}_f)-b$ for $0<a\leq\dfrac{m}{m-1}, b\geq0$\mathrm{.}Inparticular,weconsiderthreemaincasesfordifferent$a$and$b$withorwithoutconditionson$\lambda_1(\Delta_f)$. These results are extensions of Dung and Vieira, and weighted generalizations of Li-Wang, Dung-Sung and Vieira.