Positive Harmonic Functions on Complete Manifolds of Negative Curvature

Abstract

This paper studies the interaction between the geometry of complete Riemannian manifolds of negative curvature and some aspects of function theory on these spaces. The study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction; we recall some aspects of this below. There is a well-known representation of positive harmonic functions on the unit disc U. due to Herglotz [13], in terms of positive Borel measures tt on the circle S1: