Geometric implications of fast volume growth and capacity estimates

Abstract: We obtain connectivity of annuli for a volume doubling metric measure Dirichlet space which satisfies a Poincar\'e inequality, a capacity estimate and a fast volume growth condition. This type of connectivity was introduced by Grigor'yan and Saloff-Coste in order to obtain stability results for Harnack inequalities and to study diffusions on manifolds with ends. As an application of our result, we obtain stability of the elliptic Harnack inequality under perturbations of the Dirichlet form with radial type weights.