Journal of the
Korean Mathematical Society JKMS

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 $\ell$-form on $M$ is trivial if $N$ has pure curvature tensor and $M$ satisfies some geometric conditions. Then, we obtain a vanishing theorem on Riemannian manifolds with a 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 the Ricci curvature has suitable bound.

Keywords: $p$-harmonic functions, flat normal bundle, locally conformally flat, weighted $p$-Laplacian, weighted Poincare inequality

MSC numbers: 53C24, 53C21