Martin compactification of a complete surface with negative curvature

Abstract

In this paper we consider the Martin compactification, associated with the operator $$\mathcal {L}= \Delta -1$$ , of a complete non-compact surface $$\Sigma ^2$$ with negative curvature. In particular, we investigate positive eigenfunctions with eigenvalue one of the Laplace operator $$\Delta $$ of $$\Sigma ^2$$ , and prove a uniqueness result: such eigenfunctions are unique up to a positive multiplicative constant, 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 (Cao and He, Infinitesimal rigidity of steady gradient Ricci soliton in dimension three. arXiv:1412.2714 , 2014, preprint), 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.