In this talk, we shall discuss the Martin compactification of a certain non-compact surface whose curvature is negative and bounded from below but approaches zero at the infinity. We also prove a certain uniqueness result of positive eigenfunctions, with eigenvalue one, of the Laplace operator of the surface. The uniqueness problem arises in our study of deformations of three-dimensional collapsed gradient steady Ricci solitons. This is a joint work with Chenxu He at UC Riverside.

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