Vanishing results on weighted manifolds with lower bounds of the curvature operator

In this paper, we apply a new Bochner technique introduced in the recent work by Petersen and Wink to investigate vanishing properties of p -harmonic ℓ -forms on Riemannian manifolds. Assuming that M is a complete, noncompact n -dimensional manifold with an ( n − ℓ ) -positive curvature operator, we demonstrate that any p -harmonic ℓ -forms on M with finite L q -energy must be trivial. To establish this result, we consider a general framework for a complete noncompact weighted Riemannian manifold ( M , g , e − f d μ ) where the weighted curvature operator is bounded from below. By assuming the validity of a Sobolev inequality on ( M , g , e − f d μ ) , we apply the Moser iteration technique to estimate the sup-norm of forms and verify their vanishing properties.