Vanishing properties of $p$-harmonic $\mathrm{\ell }$-forms on Riemannian manifolds

In this paper, we show several vanishing type theorems for $p$-harmonic $\mathrm{\ell }$-forms on Riemannian manifolds ($p\ge 2$). First of all, we consider complete non-compact immersed submanifolds $M^n$ of $N^{n+m}$ with flat normal bundle, we prove that any $p$-harmonic $\mathrm{\ell }$-forms on $M$ is trivial if $N$ has pure curvature tensor and $M$ satisfies some geometric condition. Then, we obtain a vanishing theorem on Riemannian manifolds with weighted Poincar'{e} inequality. Final, we investigate complete simply connected, locally conformally flat Riemannian manifolds $M$ and point out that there is no nontrivial $p$-harmonic $\mathrm{\ell }$-form on $M$ provided that $\mathrm{Ric}$ has suitable bound.