Stability of p‐parabolicity under quasi‐isometries

Abstract Kanai proved the stability under quasi‐isometries of numerous global properties (including existence of Green's function, i.e., non‐parabolicity) between Riemannian manifolds of bounded geometry. Unfortunately, Kanai's hypotheses are not usually satisfied in the context of Riemann surfaces endowed with the Poincaré metric. In this work we prove the stability of p ‐parabolicity (with ) by quasi‐isometries, under hypotheses that many Riemann surfaces satisfy. Consequences for the stability of the Liouville property are obtained. In order to get our results, it is shown that each Riemannian surface with pinched negative curvature is bilipschitz equivalent to a surface with constant negative curvature.