Rough isometry and Dirichlet finite harmonic functions on Riemannian manifolds
type: Item
instance of: scholarly article
title: Rough isometry and Dirichlet finite harmonic functions on Riemannian manifolds
zbMATH Open document ID: 0934.31005
author: Yong Hah Lee
published in: Manuscripta Mathematica
publication date: 2000-01-27T00:00:00Z
review text: Let \(M\) be a Riemannian manifold. We say that \(M\) satisfies a \(D\)-Liouville property if every harmonic function with finite Dirichlet integral on \(M\) is constant. In the present paper, the author considers a generalized \(D\)-Liouville property. To be precise, the author proves that the dimension of harmonic functions with finite Dirichlet integral is invariant under rough isometries between Riemannian manifolds satisfying some local conditions. One should remark that the finiteness of the Dirichlet integral is necessary in proving the bi-Lipschitz invariance of the Liouville property. For example, see \textit{A. Grigor'yan} [Math. USSR, Sb. 60, No. 2, 485-504 (1988); translation from Mat. Sb., Nov. Ser. 132(174), No. 4, 496-516 (1987; Zbl 0646.31009)] and \textit{T. Lyons} [J. Differ. Geom. 26, 33-66 (1987; Zbl 0599.60011)]. The main result of the present paper directly generalizes those of \textit{M. Kanai} [J. Math. Soc. Japan 38, 227-238 (1986; Zbl 0577.53031)], \textit{A. Grigor'yan} (loc. cit.) and \textit{I. Holopainen} [Rev. Mat. Iberoam. 10, No. 1, 143-176 (1994; Zbl 0797.31008)]. The author also proves that the dimension of harmonic functions with finite Dirichlet integral is preserved under the rough isometries between a Riemannian manifold satisfying the same local conditions and a graph of bounded degree; and between graphs of bounded degree. These results generalize those of \textit{I. Holopainen} and \textit{P. M. Soardi} [Manuscr. Math. 94, No. 1, 95-110 (1997; Zbl 0898.31007)], and of \textit{P. M. Soardi} [Proc. Am. Math. Soc. 119, 1239-1248 (1993; Zbl 0801.31002)], respectively.
Mathematics Subject Classification ID: 31C05
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 31C20
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 58J32
zbMATH DE Number: 1319557
zbMATH Keywords: \(D\)-Liouville property
zbMATH Keywords: Royden's decomposition
zbMATH Keywords: harmonic boundary
zbMATH Keywords: extremal length
zbMATH Keywords: Poincaré inequality
zbMATH Keywords: Sobolev inequality
MaRDI profile type: Publication
Wikidata QID: Q115388421
full work available at URL: s002290050175
OpenAlex ID: W2005142804
DOI: 10.1007/S002290050175
description: scientific article; zbMATH DE number 1319557