  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.
