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 $\Gamma$.