Brownian motion and the Dirichlet problem at infinity on two-dimensional Cartan-Hadamard manifolds

After recalling the Dirichlet problem at infinity on a Cartan-Hadamard manifold, we discuss what is known and the difference between the two-dimensional and higher-dimensional cases. Turning our attention to the two-dimensional case, we prove that the Dirichlet problem at infinity on a two-dimensional Cartan-Hadamard manifold is solvable under the curvature condition $K\leq (1+\epsilon)/(r^2 \log r)$, outside of a compact set, for some $\epsilon>0$ in polar coordinates around some pole. This condition on the curvature is sharp, and improves upon the previously known case of quadratic curvature decay. Finally, we briefly discuss the issues which arise in trying to extend this method to higher dimensions.