$L_f^p$ harmonic 1-forms on complete non-compact smooth metric measure spaces

This paper studies complete non-compact smooth metric measure space $(M^n,g,\mathrm{e}^{-f}\mathrm{d}v)$ with positive first spectrum $λ_1(Δ_f)$ or satisfying a weighted Poincaré inequality with weight function $ρ$. We establish two splitting and vanishing theorems for $L_f^p$ harmonic $1$-forms under the assumption that $m$-Bakry-Émery Ricci curvature $\mathrm{Ric}_{m,n}\geq -aλ_1(Δ_f)$ or $\mathrm{Ric}_{m,n}\geq -aρ-b$ for particular constants $a$ and $b>0$. These results are inspired by the work of Han-Lin and are $L_f^p$ generalizations of previous works by Dung-Sung and Vieira for $L^2$ harmonic $1$-forms.

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