Weighted Poincar\'e inequality and rigidity of complete manifolds

Abstract

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.