  We develop Green's function estimate for manifolds satisfying a weighted
Poincare inequality together with a compatible lower bound on the Ricci
curvature. The estimate is then applied to establish existence and sharp
estimates of the solution to the Poisson equation on such manifolds. As an
application, a Liouville property for finite energy holomorphic functions is
proven on a class of complete K\"ahler manifolds. Consequently, such K\"ahler
manifolds must be connected at infinity.
