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

TL;DR

This paper investigates the solvability of the Dirichlet problem at infinity on two-dimensional Cartan-Hadamard manifolds, establishing sharp curvature conditions for solutions and discussing extensions to higher dimensions.

Contribution

It proves the Dirichlet problem at infinity is solvable under a sharp curvature decay condition in two dimensions, improving previous quadratic decay results.

Findings

Solvability of the Dirichlet problem at infinity under curvature $K extless (1+\epsilon)/(r^2 \log r)$

Curvature condition is sharp for two-dimensional Cartan-Hadamard manifolds

Discussion on extending methods to higher dimensions

Abstract

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.