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

Abstract

In this paper, we show several vanishing type theorems for $p$-harmonic $\ell$-forms on Riemannian manifolds ($p\geq2$). 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 $\ell$-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 $\ell$-form on $M$ provided that $\operatorname{Ric}$ has suitable bound.