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 ($p\ge 2$). 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 $\ell$-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 Poincaré 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 $\mathrm{Ric}$ has suitable bound.

21 pages. Comments are welcome. This is a version of our paper subbmited to a journal, last February, 2016

References in corpus (2)

• L^2 harmonics forms on non compact manifolds
• Local and global sharp gradient estimates for weighted $p$-harmonic functions