Finiteness and vanishing results on weighted Poincare inequality of complete manifolds

Abstract

We study manifolds satisfying a weighed Poincare inequality, which was first introduced by Li-Wang. We generalized one of their results by relaxing the Ricci curvature bound condition only being satisfied outside a compact set and established a finitely many ends result. We proved a vanishing result for $L^2$ harmonic 1-form provided that the weight function $\rho$ is of sub-quadratic growth of the distance function.