Vanishing theorems for p-harmonic forms on Riemannian manifolds

Abstract

In this paper, we establish several vanishing type theorems for a p-harmonic ℓ-form ω on a Riemannian manifold M. We give some vanishing theorems on a manifold with a weighted Poincaré inequality. To do so, we need to employ a Bochner formula on both ω and ⋆ω, where ⋆ is the Hodge star operator. Then, we need to control the nonlinearity of the curvature terms and inner product of forms. Moreover, using these techniques we can show that if a pure curvature tensor condition is satisfied, then we still obtain another vanishing theorem. Finally, by using Sobolev inequality in terms of Yamabe invariant, a vanishing property for p-harmonic ℓ-forms on Riemannian manifolds is also given.