On the Dirichlet Problem at Infinity and Poisson Boundary for Certain Manifolds without Conjugate 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 $\Gamma$.