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

Abstract

This paper studies complete non-compact smooth metric measure space $(M^n,g,\mathrm{e}^{-f}\mathrm{d}v)$ with positive first spectrum $\lambda_1(\Delta_f)$ or satisfying a weighted Poincar\'e inequality with weight function $\rho$. We establish two splitting and vanishing theorems for $L_f^p$ harmonic $1$-forms under the assumption that $m$-Bakry-\'Emery Ricci curvature $\mathrm{Ric}_{m,n}\geq -a\lambda_1(\Delta_f)$ or $\mathrm{Ric}_{m,n}\geq -a\rho-b$ 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.