Martin compactification of a complete surface with negative curvature

TL;DR

This paper studies the Martin compactification of negatively curved surfaces, proving a uniqueness result for positive eigenfunctions with eigenvalue one under specific boundary conditions, which aids in rigidity results for Ricci solitons.

Contribution

It establishes a uniqueness theorem for eigenfunctions of the Laplace operator on negatively curved surfaces with boundary conditions, advancing understanding of geometric analysis on such surfaces.

Findings

Unique positive eigenfunctions vanish on certain boundary parts.

Eigenfunctions satisfy growth estimates on other boundary parts.

Results support rigidity theorems in Ricci soliton theory.

Abstract

In this paper we consider the Martin compactification, associated with the operator $\mathcal{L} = \Delta -1$, of a complete non-compact surface $(\Sigma^2, ds^2)$ 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.