2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/123891We prove structure theorems for complete manifolds satisfying both the Ricci curvature lower bound and the weighted Poincaré 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 Poincaré inequality, and yields a criterion of parabolicity of connected components at infinity in terms of the weight function.Differential Geometry58J90text