Dirichlet finite harmonic functions and points at infinity of graphs and manifolds

Abstract

In this paper, we consider the Royden compactifications relative to $p$-Dirichlet integrals of infinite graphs and noncompact Riemannian manifolds, and study the behavior of rough isometries in the compactifications, proving bijective correspondence of the spaces of $p$-harmonic functions with finite $p$-energy.