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

Convergence of metric graphs and energy forms

“…Then the limit of the forms as t → +∞ also gives a resistance form on Γ. (ii) If we restrict ourselves to a class of connected, infinite graphs G = (V, E) with bounded degree, the conditions in Proposition 4.1 and Theorem 7.11 for the resistance forms E G of the graphs are invariant under quasi-isometries, after the result in [24] mentioned in the introduction.…”

“…Besides the relation to problems on convergence of networks, we are interested in the space of harmonic functions of finite Dirichlet sums on an infinite network. When we restrict ourselves to a class of connected, infinite graphs of bounded degrees, it is proved in [24] that this space is invariant under quasi-isometries (cf. Remark 1.2) in the sense that quasi-isometries between two connected infinite graphs of bounded degrees canonically induce bounded linear isomorphisms between the spaces of harmonic functions of finite Dirichlet sums; this is true for quasi-isometries between a connected infinite graph of bounded degrees and a connected, complete, noncompact Riemannian manifold such that the Ricci curvature is bounded from below and the volume of every ball of radius one is bounded from below by a positive constant.…”

Functions with finite Dirichlet sum of orderpand quasi-monomorphisms of infinite graphs

“…induces a homeomorphism between the Royden p-boundary of G and that of M which sends the harmonic p-boundary of G to that of M (see [18] for details).…”

“…We recall here a result in [18]. Let HL 1,p (M ) be the space of p-harmonic functions h on M with finite p-Dirichlet integral D p (h) = M dh p dv.…”

“…There are several important quasi-isometric invariant properties, such as having a certain type of volume growth, p-parabolicity (1 < p < +∞), the existence of spectral gaps, and so on, which have been studied in many papers (see [22], [23], [20], [34], and the references therein). In addition, the space of p-harmonic functions with finite Dirichlet sum of order p on a graph in BG and the space of p-harmonic functions with finite Dirichlet integral of order p on a manifold in BG possess invariant properties under quasi-isometries, as shown in [21], [31], and [18]. For example, it is known that Euclidean space R n of dimension n is p-parabolic if and only if p ≥ n, and there exist no nonconstant p-harmonic functions with finite Dirichlet integral of order p on R n for all p > 1; on the other hand, the hyperbolic space form H n of constant curvature −1 and dimension n is not p-parabolic for all p > 1, and it admits a lot of nonconstant p-harmonic functions with finite Dirichlet integral of order p if p > n − 1 and no such functions if p ≤ n − 1.…”