Boundaries of non-compact harmonic manifolds

TL;DR

This paper investigates the boundary structures and measure properties of non-compact harmonic manifolds, establishing boundary coincidences, measure uniqueness, and dynamical behavior of geodesic flows in these spaces.

Contribution

It proves the coincidence of Martin and Busemann boundaries, shows measure uniqueness for finite volume quotients, and demonstrates topological transitivity of geodesic flows.

Findings

Martin and Busemann boundaries coincide

Unique Patterson-Sullivan measure for finite volume quotients

Geodesic flow is topologically transitive

Abstract

In this paper we consider non-compact non-flat simply connected harmonic manifolds. In particular, we show that the Martin boundary and Busemann boundary coincide for such manifolds. For any finite volume quotient we show that (up to scaling) there is a unique Patterson-Sullivan measure and this measure coincides with the harmonic measure. As an application of these results we prove that the geodesic flow on a non-flat finite volume harmonic manifold without conjugate points is topologically transitive.