Martin compactification of a complete surface with negative curvature

Abstract

Abstract. In this paper we consider the Martin compactification, associated with the operator L = ∆ − 1, of a complete non-compact surface Σ 2 with negative curvature. In particular, we investigate positive eigenfunctions with eigenvalue one of the Laplace operator ∆ of Σ 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∞(Σ 2 ) of Σ 2 where the curvature is bounded above by a negative constant, and satisfy some growth estimate on the other part of S∞(Σ 2 ) where the curvature approaches zero. This uniqueness result plays an essential role in our recent paper [CH] 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.