In this talk We discuss the Martin compactification of a special complete noncompact
surface with negative Gaussian curvature which arises in our study of infinitesimal
rigidity of three-dimensional (collapsed) steady gradient Ricci solitons. In
particular, we investigate positive eigenfunctions with eigenvalue one of the
Laplace operator and prove a uniqueness result: such eigenfunctions are unique up to
a positive constant multiple if certain boundary behavior is satisfied. This
uniqueness result was used to prove an infinitesimal rigidity theorem for
deformations of certain three-dimensional collapsed gradient steady Ricci soliton
with a non-trivial Killing vector field. It is a joint work with Huai-Dong Cao.