Martin compactification of a complete surface with negative curvature

In this paper we consider the Martin compactification, associated with the operator $\mathcal{L} = \Delta -1$, of a complete non-compact surface $(\Sigma^2, ds^2)$ with negative curvature. In particular, we investigate positive eigenfunctions with eigenvalue one of the Laplace operator $\Delta$ of $(\Sigma^2, ds^2)$ and prove a uniqueness result: such eigenfunctions are unique up to a positive constant multiple if they vanish on the part of the geometric boundary $S_\infty(\Sigma^2)$ of $\Sigma^2$ where the curvature is bounded above by a negative constant, and satisfy some growth estimate on the other part of $S_\infty(\Sigma^2)$ where the curvature approaches zero. This uniqueness result plays an essential role in our recent paper "Infinitesimal rigidity of steady gradient Ricci soliton in three dimension" in which we prove an infinitesimal rigidity theorem for deformations of certain three-dimensional collapsed gradient steady Ricci soliton with a non-trivial Killing vector field.