Potential Theory on Gromov Hyperbolic Manifolds of Bounded Geometry

Abstract

In the 1980s Alano Ancona developed a profound potential theory on Gromov hyperbolic manifolds of bounded geometry. Since then, such hyperbolic spaces have become basic in geometry, topology and group theory. In this paper we make Ancona's original work, addressed to a rather advanced audience, approachable for a wider readership without firm background in potential theory. We streamline the arguments to make the results depend only on coarse geometro-analytic invariants and we derive useful details not found in the original sources. We explain remarkable relations of these results to recent developments in the potential theory and quasi-conformal geometry of Euclidean domains.