The $p$-harmonic boundary for quasi-isometric graphs and manifolds

TL;DR

This paper establishes a homeomorphism between the $p$-harmonic boundaries of quasi-isometric graphs and manifolds, and shows a correspondence between their $p$-harmonic functions, bridging discrete and continuous geometric analysis.

Contribution

It proves the homeomorphism of $p$-harmonic boundaries and a bijection of $p$-harmonic functions for quasi-isometric graphs and manifolds, extending harmonic analysis to a broader geometric context.

Findings

$p$-harmonic boundary of $G$ is homeomorphic to that of $M$

Bijection between $p$-harmonic functions on $G$ and $M$

Results hold under bounded degree and specific properties of $M$

Abstract

Let $p$ be a real number greater number greater than one. Suppose that a graph $G$ of bounded degree is quasi-isometric with a Riemannian manifold $M$ with certain properties. Under these conditions we will show that the $p$-harmonic boundary of $G$ is homeomorphic to the $p$-harmonic boundary of $M$. We will also prove that there is a bijection between the $p$-harmonic functions on $G$ and the $p$-harmonic functions on $M$.