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

Abstract

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 $\ell$-forms on Riemannian manifolds. Assuming that $M$ is a complete, noncompact $n$-dimensional manifold with an $(n-\ell )$-positive curvature operator, we demonstrate that any $p$-harmonic $\ell$-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 $(M,g,e^{-f}d\mu )$ where the weighted curvature operator is bounded from below. By assuming the validity of a Sobolev inequality on $(M,g,e^{-f}d\mu )$, we apply the Moser iteration technique to estimate the sup-norm of forms and verify their vanishing properties.