Vanishing properties of $f$-minimal hypersurfaces in a complete smooth metric measure space

Let $(N^{n+1},g,e^{-f}dv)$ be a complete smooth metric measure space with $M^{n}$ being a complete noncompact $f$-minimal hypersurface in $N^{n+1}$. In this paper, we extend the classical vanishing theorems for $L^2$-harmonic $1$-forms on a complete minimal hypersurface to a weighted manifold. In addition, we obtain a vanishing result under the assumption that $M^n$ has sufficiently small weighted $L^n$-norm of the second fundamental form on $M^{n}$, which can be regarded as a generalization of a result by Yun and Seo.
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