Geometric and Martin boundaries of a Cartan-Hadamard surface

TL;DR

This paper establishes criteria for solving the Dirichlet problem at infinity on Cartan-Hadamard surfaces, linking curvature bounds to boundary identifications and Martin boundary properties.

Contribution

It provides new curvature bounds ensuring DPI solvability and clarifies the relationship between Martin and geometric boundaries on these surfaces.

Findings

Radial curvature bounds imply DPI solvability.

Solvability of DPI leads to a surjection from Martin to geometric boundary.

Matching curvature bounds identify Martin and geometric boundaries.

Abstract

We give a general criterion for the Dirichlet problem at infinity (DPI) on a Cartan-Hadamard surface to be solvable, which we primarily use to give the best possible upper radial radial curvature bound for solvability of the DPI, but which is also flexible enough to accommodate flats. In particular, any (upper) radial curvature bound which implies transience also implies solvability of the DPI, which is perhaps surprising. Taking advantage of the structure provided by uniformization, we show that solvability of the DPI implies there is a natural continuous surjection of the Martin boundary onto the geometric boundary at infinity. Finally, we give matched upper and lower radial curvature bounds that imply the natural identification of the geometric and Martin boundaries (for Cartan-Hadamard surfaces) that are more generous than the bounds that are known in arbitrary dimension.