Summary: We deal with complete submanifolds with weighted Poincaré inequality. By assuming the submanifold is $\delta $-stable or has sufficiently small total curvature, we establish two vanishing theorems for $L^p$ harmonic $1$-forms, which are extensions of the results of Dung-Seo and Cavalcante-Mirandola-Vitório.
 Miyaoka, R.: $L^2$ harmonic $1$-forms on a complete stable minimal hypersurface. Geometry and Global Analysis T. Kotake et al. Int. Research Inst., Sendai 1993, Tôhoku Univ., Mathematical Institute (1993), 289-293. MR 1361194 | Zbl 0912.53042