Graphs with nonnegative curvature outside a finite subset, harmonic functions and number of ends

TL;DR

This paper investigates graphs with nonnegative curvature outside a finite set, demonstrating that such graphs have a finite-dimensional space of bounded harmonic functions and finitely many non-parabolic ends, using discrete Gromov-Hausdorff convergence.

Contribution

It introduces a novel approach using discrete Gromov-Hausdorff convergence to analyze harmonic functions and ends in graphs with curvature constraints outside finite sets.

Findings

Finite-dimensional space of bounded harmonic functions

Finitely many non-parabolic ends

Application of discrete Gromov-Hausdorff convergence

Abstract

We study graphs with nonnegative Bakry-\'Emery curvature or Ollivier curvature outside a finite subset. For such a graph, via introducing the discrete Gromov-Hausdorff convergence we prove that the space of bounded harmonic functions is finite dimensional, and as a corollary the number of non-parabolic ends is finite.