On Doyle-Grigor'yan criterion for non-parabolicity (Q6498402)

A sufficient condition is given by Doyle and Grigor'yan for the non-parabolicity of a complete Riemannian manifold, and Grigor'yan (1999) asked whether this is also a necessary condition. In this paper the authors provided a negative answer to this question for dimension larger than or equal to four, by constructing non-parabolic Riemannian manifolds not satisfy the Doyle-Grigor'yan condition.