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

TL;DR

This paper establishes vanishing theorems for p-harmonic orms on various classes of Riemannian manifolds, showing under what geometric and curvature conditions these forms must be trivial.

Contribution

It provides new vanishing theorems for p-harmonic orms on manifolds with specific curvature, geometric, and inequality conditions, extending previous results.

Findings

p-harmonic orms are trivial on certain non-compact submanifolds with flat normal bundle

vanishing results on manifolds with weighted Poincare9 inequality

no nontrivial p-harmonic orms on simply connected, conformally flat manifolds with bounded Ricci

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.