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

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