Weighted $L^2$ harmonic 1-forms and the topology at infinity of complete noncompact weighted manifolds

Abstract

In this paper, we prove that a complete noncompact weighted manifold supporting the weighted Sobolev inequality has at least linear weighted volume growth. We also obtain vanishing results and finiteness theorems for the weighted $L^2_f$ $f$-harmonic 1-forms on a complete noncompact weighted manifold supporting the weighted Sobolev inequality.