Weighted Poincaré inequality and rigidity of complete manifolds

Peter Li, Jiaping Wang

Published 2007-01-24, updated 2007-02-14Version 2

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.