On the Dirichlet Problem at Infinity and Poisson Boundary for Certain Manifolds without Conjugate Points

TL;DR

This paper studies the existence of bounded harmonic functions on certain non-positively curved manifolds without conjugate points, establishing boundary value problems and identifying the Poisson boundary via harmonic measures.

Contribution

It proves the existence of harmonic extensions for continuous boundary functions on these manifolds and characterizes the Poisson boundary using harmonic measures for rank 1 manifolds without focal points.

Findings

Harmonic functions extend continuously from the ideal boundary.

The harmonic measures form the Poisson boundary for the group actions.

Results apply to uniform visibility and rank 1 manifolds without focal points.

Abstract

In this paper, we investigate the problem of the existence of the bounded harmonic functions on a simply connected Riemannian manifold $\widetilde{M}$ without conjugate points, which can be compactified via the ideal boundary $\widetilde{M}(\infty)$. Let $\widetilde{M}$ be a uniform visibility manifold which satisfy the Axiom $2$, or a rank $1$ manifold without focal points, suppose that $\Gamma$ is a cocompact discrete subgroup of $Iso(\widetilde{M})$, we show that for a given continuous function on $\widetilde{M}(\infty)$, there exists a harmonic extension to $\widetilde{M}$. And furthermore, when $\widetilde{M}$ is a rank $1$ manifold without focal points, the Brownian motion defines a family of harmonic measures $\nu_{\ast}$ on $\widetilde{M}(\infty)$, we show that $(\widetilde{M}(\infty),\nu_{\ast})$ is isomorphic to the Poisson boundary of $\Gamma$.