# On the Martin boundary of rank 1 manifolds with nonpositive curvature

## TL;DR

This paper investigates the Martin boundary of rank 1 nonpositively curved manifolds with compact quotients, showing a natural identification with a subset of the geometric boundary, advancing understanding in geometric analysis.

## Contribution

It extends the description of the Martin boundary to rank 1 manifolds with nonpositive curvature, identifying a residual set with part of the boundary, addressing a problem posed by Yau.

## Key findings

- Residual set in the geometric boundary corresponds to a subset of the Martin boundary.
- Provides a partial answer to an open problem in geometric analysis.
- Adapts Ancona's argument for the nonpositive curvature setting.

## Abstract

For a manifold with nonpositive curvature, the Martin boundary is described by the behavior of normalized Green's functions at infinity. A classical result by Anderson and Schoen states that if the manifold has pinched negative curvature, the geometric boundary is the same as the Martin boundary. In this paper, we study the Martin boundary for rank 1 manifolds admitting compact quotients. It is proved that a residual set in the geometric boundary can be identified naturally with a subset of the Martin boundary. This gives a partial answer to one of the open problems in geometry collected by Yau. Our proof is a modification of an argument due to Ancona.