Brownian motion and harmonic functions on manifolds of negative curvature

We investigate positive solutions of the equation $\Delta u=0$, where $\Delta$ is the Beltrami–Laplace operator on manifold $M$ of negative curvature $K$. In section 3 we prove the existence and uniqueness of the Dirichlet problem with a continuous boundary function defined on the absolute of the manifold $M$. If the curvature $K$ changes slowly at infinity (see condition 2), we prove that the structure of the space of minimal positive solutions of $\Delta u=0$ is the same as in the case of constant negative curvature, i. e. there is a one-to-one correspondence between points of the absolute and normalized minimal positive solutions of $\Delta u=0$.