Weighted Poincar\'e inequality and rigidity of complete manifolds

We prove structure theorems for complete manifolds satisfying both the Ricci curvature lower bound and the weighted Poincar\'e inequality. In the process, a sharp decay estimate for the minimal positive Green's function is obtained. This estimate only depends on the weight function of the Poincar\'e inequality, and yields a criterion of parabolicity of connected components at infinity in terms of the weight function.